% plain TeX \def\giorno{25/11/96} \def\G{{\cal G}} \def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\F{{\cal F}} \def\I{{\cal I}} \def\L{{\cal L}} \def\M{{\cal M}} \def\R{{\cal R}} \def\U{{\cal U}} \def\V{{\cal V}} \def\X{{\cal X}} \def\Y{{\cal Y}} \def\Z{{\cal Z}} \def\s{\sigma} \def\S{\Sigma} \def\Ga{\Gamma} \def\ga{\gamma} \def\d{{\rm d}} \def\de{\delta} \def\De{\Delta} \def\eps{\varepsilon} \def\phi{\varphi} \def\om{\omega} \def\Om{\Omega} \def\La{\Lambda} \def\la{\lambda} \def\a{\alpha} \def\b{\beta} \def\pa{\partial} \def\T{{\rm T}} \def\sse{\subseteq} \def\ss{\subset} \def\<{\langle} \def\>{\rangle} \def\Fmn{F_{\mu \nu}} \def\D{\nabla} \def\boh{CF } \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\ref#1{[#1]} \def\~#1{{\widetilde #1}} \def\=#1{{\bar #1}} \def\remark{{\tt Remark.}} \def\wnote#1{\footnote{$^{***}$}{#1} } \font\petitrm = cmr9 \font\petitit = cmsl9 \font\petitbf = cmbx9 {\nopagenumbers ~ \vskip 2 truecm \centerline{\bf Commuting-flow symmetries and common solutions} \medskip \centerline{\bf to differential equations with common symmetries} \vskip 3 truecm \centerline{Giuseppe Gaeta$^*$} \medskip \centerline{\it I.H.E.S., 35 Route de Chartres} \centerline{\it 91440 Bures sur Yvette (France)} \centerline{\tt gaeta@ihes.fr} \bigskip \bigskip \centerline{Paola Morando} \medskip \centerline{\it Dipartimento di Matematica, Politecnico di Torino,} \centerline{\it Corso Duca degli Abruzzi 24, 10129 Torino (Italy)} \centerline{\tt morando@polito.it} \vskip 4 truecm {\bf Abstract:} We point out that in certain cases, all the differential equations (for given indipendent and dependent variables) possessing a given symmetry necessarily share a common solution. We characterize this situation and the common solution, and give concrete examples. \vfill \parindent=0pt {\petitrm $^*$ Permanent address: Dept. of Mathematics, Loughborough University, Loughborough LE11 3TU (Great Britain).} \footnote{}{{\tt \giorno}} \eject} \pageno=1 \parskip=10pt \parindent=0pt ~ \bigskip\bigskip {\bf Introduction} \bigskip We are all familiar with elementary occurences of a remarkable phenomenon: in some cases, {\it all} the differential equations possessing a given symmetry share a common solution. As trivial examples of this fact it suffices to think of reflection-invariant ODEs in $\R^1$, ${\dot x} = f(x,t)$ [these are identified by $f(x,t) = - f(-x,t)$], which all share the common solution $x(t) \equiv 0$; or also reflection and/or rotation invariant ODEs in $\R^n$, which again share the common (trivial) solution $x(t) \equiv 0$. The purpose of this short note is to point out that this phenomenon can be present in far less trivial cases, for both ODEs and PDEs; and to give a way to characterize the common solutions, at least for a relevant class of symmetries, i.e. the ``commuting-flow symmetries'' defined below (which are natural in a dynamical systems geometric approach). This question can be tackled making use of Michel's theory (see below), and indeed our result can be seen as a simple corollary -- maybe reaching unexpected fields -- of the original Michel theorem. A classical result of L. Michel \ref{1} ensures that, given a finite dimensional smooth manifold $M$ with an action of a compact Lie group $G$, we can identify the points $x \in M$ which are critical for {\it all} the $G$-invariant potentials $V(x)$. The investigation of Michel was motivated principally by fundamental problems in Physics (the theory of strong interactions) \ref{2}, but the theorem in itself is of a geometric nature, and of a much wider applicability to a range of problems involving {\it spontaneous symmetry breaking}. The Michel's result can be suitably generalized to variational problems in function spaces \ref{3}, and it has been recently extended to consider gauge invariant functionals \ref{4}. On the other side, the proof given by Michel in \ref{1} for his theorem immediately generalizes to consider equivariant dynamics defined by a vector field on $M$, and its fixed points; it also guarantees, more generally, the invariance under the equivariant dynamics of a family of subspaces of $M$ (identified by sharing a common isotropy subgroup for the $G$-action). In this way, one could also readily recover a number of fundamental results in equivariant dynamics and equivariant bifurcation theory, such as the reduction lemma of Golubitsky and Stewart \ref{5} and the equivariant branching lemma of Cicogna and Vanderbauwhede \ref{6}, originally obtained with no use of Michel's theory \ref{7}. It is thus quite natural to ask if, beside the extension of the variational aspects of Michel theory to infinite dimensional spaces \ref{4}, one could not obtain also a similar extension for its equivariant dynamics aspects. The purpose of this note is to show that this is indeed the case, and namely that we can identify subsets of a given function space $\F$ which are invariant under any dynamics on $\F$ which satisfy certain symmetry requirements with respect to an algebra $\G$ of vector fields; moreover, proceeding pretty much as in the finite dimensional case, we can identify functions $f_0 \in \F$ which are solutions for {\it any} differential equation $\De$ which is $\G$-invariant (in a sense to be precisely defined in the following). Our discussion makes use of ideas and formalisms from the symmetry theory of differential equations \ref{8-11}, which we assume is known -- at least in its basics -- to the reader. Also, our discussion will be conducted at a rather formal level; thus, in particular, we will assume the existence of flows generated by the (generalized) vector fields to be considered in the following. \vfill\eject {\bf 1. Evolutionary vector fields and symmetry} \bigskip Let us consider a smooth manifold $M = \X \times \Y$, where the $x \in \X \sse \R^q$ and the $u \in \Y \sse \R^p$ will be thought as, respectively, the independent and dependent variables for the differential equations to be considered in a moment. We consider a vector field $Y$ on $M$, which we write as $$ Y \ = \ \sum_{i=1}^q \xi^i (x,u) {\pa \over \pa x^i} \ + \ \sum_{\a=1}^p \phi^\a (x,u) {\pa \over \pa u^\a} \ \eqno(1) $$ (with $\xi$, $\phi$ smooth functions on $M$); as it is well known \ref{8}, to any such vector field we can associate an {\it evolutionary representative}; this will be denoted by $X$ and is given by $$ X \ = \ \[ \phi^\a (x,u) - \sum_i \xi^i (x,u) u^\a_i \] \pa_\a \ \equiv Q^\a [u] \pa_\a \ . \eqno(2) $$ Here we write $\pa_\a $ for $\pa / \pa u^\a$, and $u^\a_i$ for $\pa u^\a / \pa x^i$ (these notations will be understood throughout in the following; from now on we will also understand summation over repeated indices). The notation $Q[u]$ -- used extensively in the following -- means that $Q$ is a function of $x$, $u$, and $u$ derivatives of any order (although in the present case these are only of first order). We will also write $X$ as $X_Q$, to stress the association with the vector $Q[u]$; more generally, $X_P$ will denote the evolutionary vector field $X_P = P^\a [u] \pa_\a$. It should be recalled that, while $Y$ is a legitimate vector field on $M$, its evolutionary representative $X$ should be considered with some extra care; we can either consider it as a notation for a vector field in the function space $\M$ (the space of smooth functions from $\R^q$ to $\R^p$), such that $ \( I + \eps X_Q \) $ transforms $f(x)$ into the function $$ \~f^\a (x) = f^\a (x) + \eps \[ \phi^\a (x, f(x) ) - \xi^i (x,f(x)) f^\a_i (x) \] \ , \eqno(3) $$ either as a generalized vector field acting in the jet space associated to $M$; in this case, one should more precisely consider the infinite order prolongation $X^*$ of $X$, acting in the infinite order jet space $J^* (M)$, in order to have a proper vector field. It should also be stressed that, for the argument to be considered below, one could start directly from $X^*$, i.e. consider generalized vector fields rather than limit to consider only representative of geometric vector fields. Let us now consider a system of $p$ differential equations for functions $u^\a = u^\a (x) \in \M$; this will be written as $\De [u] = 0$ (this will be a vector $\De^\a$, with $\a = 1,...,p$; we will say for simplicity ``the equation'' $\De$ to refer to this system). With the equation $\De$ we associate an evolutionary generalized vector field $X_\De$ given by $$ X_\De \ = \ \De^\a [u] \pa_\a \ ; \eqno(4) $$ as it was the case for $X_Q$, this can be seen either as a vector field on $\M$, either as denoting the infinite prolongation $X_\De^*$ acting in $J^* (M)$. We recall \ref{8} that $X_Q$ is a {\it symmetry} of $\De$ if $$ X_Q^* (\De ) \ = \ 0 \ , \eqno(5) $$ is satisfied whenever $\De = 0$, and a {\it strong symmetry} of $\De$ if (5) is identically satisfied. \bigskip\bigskip \vfill\eject {\bf 2. \boh symmetries} \bigskip Given two evolutionary generalized vector fields (EGVF), $X_P = P^\a[u] \pa_\a$ and $X_Q = Q^\a [u] \pa_\a$, we can consider their commutator $\[ X_P , X_Q \]$; this will be again an EGVF $Y_R = R^\a [u] \pa_\a$, with $$ R^\a [u] = X_P^* \( Q^\a \) - X_Q^* \( P^\a \) \ . \eqno(6) $$ It is easy to check \ref{8} that this gives indeed a proper commutator, i.e. the usual properties of the commutator are satisfied. Given a differential equation $\De$, and an EGVF $X_Q$, we say that $X_Q$ is a {\bf Commuting Flow (CF) symmetry} for $\De$ if the associated EGVF commute (in the above sense), i.e. if $$ \[ X_Q , X_\De \] \ = \ 0 \ . \eqno(7) $$ For a given equation $\De$, we denote by $\G_\De$ the set of EGVF which commute with $X_\De$ (this is easily seen to be a Lie algebra, with $X_\De$ belonging to its center). Conversely, for a given EGVF $X_Q $ we denote by $\I_Q$ the set of differential equations which have $X_Q$ as a \boh symmetry; again, the $X_\De$ corresponding to $\De \in \I_Q$ form a Lie algebra. {\bf Lemma 1.} {\it If $X_Q$ is a \boh symmetry for $\De$, then it is also a symmetry of $\De$ in the ordinary sense.} {\tt Proof.} Indeed, on $\De =0$ we have that both $X_\De$ and $X_\De^*$ vanish, as it is clear since $X_\De = \De^\a [u] \pa_\a$ and $X_\De^* = \sum_J (D_J \De^\a [u] ) (\pa / \pa u^\a_J )$ (here $J$ is again a multiindex). Thus, $X_\De^* (Q) = 0$ on $\De=0$ for any $Q$, and the vanishing of $\[ X_Q , X_\De \] $ is equivalent to the vanishing of $X_Q^* ( \De )$, which is the condition (4) for $X_Q$ to be a symmetry of $\De$ in the ordinary sense. $\odot$ \bigskip\bigskip {\bf 3. Invariance of fixed space and common solutions to symmetric equations} \bigskip With any $X_Q : \F \to \T \F$ we associate the set $F(Q) \sse \F$ of functions which are fixed points for the vector field $X_Q$; equivalently, this means that $Q[f] = 0$ for $f \in F(Q)$. {\bf Lemma 2.} {\it If $X_Q$ is a \boh symmetry of $\De$, then $X_\De$ leaves $F(Q)$ globally invariant, i.e. $X_\De : F(Q) \to \T F(Q)$.} {\tt Proof.} Let us consider, for the sake of clarity, the integrated version of the commutation relations (assuming suitable existence and unicity conditions for the flows of the vector fields); we have -- with $\la$, $\mu$ real parameters -- that $$ e^{\la Q[u]} e^{\mu \De [u]} (f) \ = \ e^{\mu \De [u]} e^{\la Q[u]} (f) \ ; \eqno(8) $$ from $f \in F(Q)$ it follows that $e^{\la Q} (f) = f$, and thus we have that $$ e^{\mu \De [u] } \( f \) \in F(Q) \quad \forall \mu \ , \eqno(9) $$ from which the lemma follows immediately. $\odot$ {\bf Corollary.} {\it If there exists an $f_0 \in F(Q)$ isolated in $F(Q)$, then necessarily $f_0$ is solution to any $\De \in \I_Q$.} {\tt Proof.} In this case, $f_0$ is a fixed point of $\De [u]$; but this is equivalent to being a solution to $\De$. $\odot$ \remark When we speak of an ``isolated'' function, we are implicitely assuming $\M$ is equipped with some natural topology; in the examples below, in which we consider a space of $L^2$ functions, this will be the $L^2$ norm. Actually, we could consider any topology in which the flows of $X_Q$ and $X_\De$ are continuous. The above lemma and corollary are immediately generalized to the case of a whole algebra of \boh symmetries; we denote by $\G \equiv \G_\De$ this (possibly, but not necessarily, abelian) algebra, and say that $\G$ is a \boh symmetry algebra for $\De$ if $[X_Q , X_\De ] = 0$ for any $X_Q \in \G$. In this case we denote by $F (\G )$ the set of functions invariant under all the $X_Q \in \G$, and by $\I_\G$ the set of equations for which $\G$ is a \boh symmetry algebra. We have then immediately the {\bf Lemma 3.} {\it If $\G$ is a \boh symmetry algebra of $\De$, then $X_\De$ leaves $F(\G )$ globally invariant, i.e. $X_\De : F(\G ) \to \T F(\G )$.} {\bf Corollary.} {\it If there exists a $f_0 \in F(\G )$ isolated in $F(\G )$, then necessarily $f_0$ is solution of any $\De \in \I_\G$.} It should be noted that it is not easy to have such an isolated $f_0 \in F(Q) \sse \M$; e.g., in the case where $Q[u]$ is a homogeneous linear function of $u$, we have that necessarily $F(Q)$ is a linear subspace of $\M$, and we cannot have isolated points (unless $F[Q]$ reduces to the zero function alone). However, we have so far considered the whole function space $\M$; when we deal with differential equations, we usually have to look for solutions in some given function space, less general than $\M$; restricting to a smaller function space makes it easier to find isolated fixed points $f_0$. Let us consider a closed subspace $\F \sse \M$, which we think as fixed once and for all. We have now to consider only those $\De$ such that $X_\De$ are vector fields on $\F$. We can then obtain again the equivalent of the lemma 3 and its corollary given above, provided we consider now the intersection of $F(\G ) \sse \M$ with $\F$. We summarize our discussion in the following form. {\bf Proposition.} {\it Let $\G$ be an algebra of EGVF on $\M$, and $F(\G )$ the space of fixed points for $\G$; let $\F \sse \M$ be a subspace of $\M$, such that there exists a $f_0$ isolated in $F(\G ) \cap \F$. Then, any differential equation $\De \in \I_\G$ such that $X_\De : \F \to \T \F$ admits $f_0 (x) \in \F$ as a solution.} \bigskip \bigskip {\bf 4. Examples} \bigskip {\it Example 1.} Let us consider $\X = S^1$, $\Y = R$, so that $\M$ is the space of $\C^\infty$ function $f:S^1 \to R$; in this we select the space $\F$ of functions which satisfy $$ ||f||^2 \ \equiv \ {1 \over 2 \pi} \int_{0}^{2 \pi} | f(x) |^2 \d x \ = \ 1 \ . \eqno(10) $$ We consider $Y = - \pa / \pa x$, and correspondingly $$ X_Q = u_x \pa_u \quad ; \quad Q[u] = u_x \eqno(11) $$ Let us now look for $F(Q) \cap \F$; $F(Q)$ is given by constant functions $f(x) = c$, and the constraint (10) selects the two functions $ f_\pm (x) = \pm 1 $. These are obviously isolated in $F(Q) \cap \F$. Let us now determine the $\De \in \I_Q$; the condition $\[ X_Q , X_\De \] = 0$ amounts to $\pa \De / \pa x = 0$. Indeed, $X_Q^* = \sum_j (D_j u_x ) \pa_{u_j}$, while the first prolongation of $X_\De$ is given by $X_\De^{(1)} = \De \pa_u + (D_x \De ) \pa_{u_x}$. Thus, $$ X_Q^* (\De ) - X_\De^* (Q) = \sum_j (D_j u_x ) {\pa \De \over \pa u_j} - D_x \De \ . \eqno(12) $$ Expanding $D_x \De = \De_x + \sum_j (\pa \De / \pa u_j ) u_{x,j}$ (where $u_{x,j} = D_j u_x$), we get that (12) reduces to $\De_x$. Thus, $\De \in \I_Q$ are written as $$ \De \ = \ \De \( u, u_x , u_{xx}, ... \) = 0 $$ Now we have to check that all such $\De$ which moreover have $X_\De$ leaving $\F$ invariant admit $f_\pm (x)$ as solutions. The requirement $X_\De : \F \to \F$ amounts to asking that $$ || e^{\la X_\De} (f) || = 1 \eqno(13) $$ whenever $|| f|| = 1$. In the limit $\la \to 0^+$, $$ e^{\eps X_\De} (f) \ \simeq \ \( I + \eps X_\De \) (f) \simeq f + \eps \De [f] \ , $$ and thus $$ || e^{\la X_\De} (f) ||^2 = || f ||^2 + 2 \eps \< f \cdot \De [f] \> + o (\eps ) \ . \eqno(14) $$ As (13) must hold for any $f \in \F$, it must in particular hold on $f_\pm$; evaluating (14) in $f_\pm$ we obtain then $\De ( \pm 1 , 0 , 0 , 0 , ... ) = 0$, i.e. that $f_\pm$ is a solution to $\De$. \bigskip {\it Example 2.} We consider now an example with $u=u(x_1,x_2)$ and an algebra of \boh symmetries spanned by two vector fields, i.e. $$ \eqalign{ Y_1 \ =& \ - x_1 \pa_{x_2} + x_2 \pa_{x_1} = - \pa_\theta \cr Y_2 \ =& \ x_1 \pa_{x_1} + x_2 \pa_{x_2} + k u \pa_u = r \pa_r + ku \pa_u \ , \cr} \eqno(15) $$ where $(r,\theta )$ are polar coordinates in the $(x_1,x_2)$ plane; clearly, $Y_1$ represents a rotation in the independent variables, and $Y_2$ a scale transformation. Correspondingly, we have $$ X_1 = u_\theta \pa_u \quad , \quad X_2 = (k u - u u_r ) \pa_u \ . \eqno(16) $$ and it is easy to see that $$ F (Q_1 ) = \{ u=u(r) \} \quad , \quad F (Q_2 ) = \{ u(x_1 , x_2 ) = \sum_{b=0}^k c_b x_1^b x_2^{k-b} \} \eqno(17) $$ (with $c_b$ real constants) and therefore $$ F(\G ) \ = \ \{ u = \a r^k \} \qquad (\a \in \R ) \ . \eqno(18) $$ We will consider, for the sake of clarity, only first order polynomial PDEs, i.e. we assume $$ \De \ = \ \De \( r , \theta ; u , u_r , u_\theta \) \ ; \eqno(19) $$ this also means that $$ X_\De^{(1)} \ = \ \De \pa_u + \( D_r \De \) \pa_{u_r} + \( D_\theta \De \) \pa{u_\theta} \eqno(20) $$ (in the following computations, we only need this first order prolongation of $X_\De$). With this, it follows from straightforward computations that $$ \eqalignno{ \[ X_1 , X_\De \] \ =& \ - {\pa \De \over \pa \theta} & (21) \cr \[ X_2 , X_\De \] \ =& \ - k \De + r {\pa \De \over \pa r} + k u {\pa \De \over \pa u} + k u_\theta {\pa \De \over \pa u_\theta} + (k-1) u_r {\pa \De \over \pa u_r} & (22) \cr} $$ The (21), and the polynomial assumption on $\De$, guarantees that $\De \in \I_{Q_1}$ implies $$ \De \ = \ \ga_{abcd} r^a u^b u_r^c u_\theta^d \ ; \eqno(23) $$ moreover, (22) requires that, in order to have $\De \in \I_\G$, only the $\ga_{abcd}$ with $$ k \ (b+c+d-1) \ = \ c-a \eqno(24) $$ can be nonzero. We can now check that indeed $X_\De$ leaves the space of functions of the form $u = \a r^k $ invariant. In facts, $$ X_\De (\a r^k ) \ = \ \ga_{abcd} r^a \a^b r^{kb} (\a k)^c r^{c(k-1)} \delta_{d,0} \ = \ \b \ga_{abc0} r^{a + kb + c(k-1)} \ , \eqno(25) $$ and it follows from (24) that the exponent of $r$ in the final expression is just $k$. This shows that indeed $X_\De : F(\G ) \to \T F(\G )$, as claimed. Let us now consider again $\F$ given by the functions of unit norm, where we define $$ || f ||^2 = \< f , f \> \equiv \int_D | f(x_1 , x_2 ) |^2 \d x_1 \d x_2 \eqno(26) $$ (here $D$ is the unit disk $r \le 1$); in this case the only functions of $F(\G )$ which are also in $\F$ are $$ f_\pm (x_1 , x_2 ) = \pm \a_0 r^k \qquad [\a_0 = \sqrt{(k+1)/\pi} ] \ . \eqno(27) $$ We rewrite the action of $X_\De$ as $$ \( I + \eps X_\De \) (f) = f + \eps \De [f] \ ; \eqno(28) $$ thus -- for $||f||=1$ -- the norm of this is one, i.e. $X_\De $ leaves $\F$ invariant, if and only if $$ \< f , \De [f] \> \ = \ 0 \ . \eqno(29) $$ We notice that $F(\G )$ is a one dimensional linear space, so that $X_\De : F(\G ) \to \T F(\G )$ means that necessarily $X_\De (f) = \la f$, for some real number $\la$, for any $f \in F( \G )$. With this, (29) means $$ \< f , \la f \> \equiv \la ||f||^2 = 0 \ , \eqno(30) $$ which can be true only for $\la = 0$. This shows that the condition that $X_\De : \F \to \T \F$ guarantees also that the isolated points in $F(\G ) \cap \F$ are solutions to $\De$, as claimed by our theorem. \bigskip {\it Example 3.} Finally, let us consider $\X = R \times R_+$ and $\Y = R^1$; then $\M$ is the space of $\C^\infty$ functions $f : R^2 \to R$, and in this we select the space $\F$ of functions which satisfy, with $(x,t)$ coordinates in $\X$, $$ \int_{- \infty}^{+ \infty} f(x,t) \d x \ = \ 1 \ \eqno(31) $$ and such that $f_x(x,t)$ go to zero when $x$ go to infinity. We consider $Y = 2 t \pa_x - xu \pa_u$, and correspondingly $$ X \ = \ - \( xu + 2 t u_x \) \pa_u \ ;\eqno(32) $$ notice that $Y$ is one of the Lie-point symmetries of the heat equation \ref{8-11}, and it can be checked that $X$ is a \boh symmetry of it. It can also be checked easily that the functions in $F(Q)$ are those of the form $$ f(x,t) \ = \ k(t) \ e^{- x^2 /4 t} \ , \eqno(33) $$ and the intersection of this $F(Q)$ with $\F$ yields precisely $$ f_0 (x,t) \ = \ {1 \over \sqrt{4 \pi t} } e^{- x^2 / 4t } \ , \eqno(34) $$ i.e. the fundamental solution of the heat equation. \vfill\eject ~ \bigskip\bigskip {\bf References} \bigskip\bigskip \def\CMP{{\petitit Comm. Math. Phys.}} \def\ref#1{\item{[#1]} } \baselineskip=10pt \def\tit#1{{``#1''}} \ref{1} L. Michel, \tit{Points critiques de fonctions invariantes sur une G-vari\'et\'e}, {\it Comptes Rendus Acad. Sci. Paris} {\petitbf 272-A} (1971), 433-436 \ref{2} L. Michel and L. Radicati, \tit{Properties of the breaking of hadronic internal symmetry}, {\it Ann. Phys. (N.Y.)} {\petitbf 66} (1971), 758; \tit{The geometry of the octet}, {\it Ann. I.H.P.} {\petitbf 18} (1973), 185 \ref{3} R.S. Palais, \tit{The principle of symmetric criticality}, {\it Comm. Math. Phys.} {\petitbf 69} (1979), 19-30 \ref{4} G. Gaeta and P. Morando, \tit{Michel theory of symmetry breaking and pure gauge theories}, preprint 1996 \ref{5} M. Golubitsky and I.N. Stewart, \tit{Hopf bifurcation in the presence of symmetry}, {\it Arch. Rat. Mech. Anal.} {\petitbf 87} (1985), 107-165; M. Golubitsky, D. Schaeffer and I. Stewart, {\it Singularities and groups in bifurcation theory - vol. II}, Springer (New York) 1988 \ref{6} G. Cicogna, \tit{Symmetry breakdown from bifurcation}, {\petitit Lett. Nuovo Cimento} {\bf 31} (1981), 600-602; and \tit{A nonlinear version of the equivariant bifurcation lemma}, {\petitit J. Phys. A} {\bf 23} (1990), L1339-L1343; A. Vanderbauwhede, {\petitit Local bifurcation and symmetry}, Pitman (Boston) 1982 \ref{7} G.Gaeta, \tit{A splitting lemma for equivariant dynamics}, {\it Lett. Math. Phys.} {\petitbf 33} (1995), 313-320; \tit{Splitting equivariant dynamics}, {\it Nuovo Cimento B} {\petitbf 110} (1995), 1213-1226 \ref{8} P.J. Olver, {\it Applications of Lie groups to differential equations}, Springer (New York) 1986 \ref{9} G.W. Bluman and S. Kumei, {\petitit Symmetries and differential equations}, Springer (New York) 1989 \ref{10} H. Stephani, {\it Differential equations. Their solution using symmetries}, Cambridge University Press (Cambridge UK) 1989 \ref{11} G. Gaeta, {\petitit Nonlinear symmetries and nonlinear equations}, Kluwer (Dordrecht) 1994 \bye