  The normal form for a system of ode's is constructed from its polynomial
symmetries of the linear part of the system, which is assumed to be
semi-simple. The symmetries are shown to have a simple structure such as
invariant function times symmetries of degree one called basic symmetries. We
also show that the set of symmetries naturally forms an infinite dimensional
Lie algebra graded by the degree of invariant polynomials. This implies that if
this algebra is non-commutative then the method of multiple scales with more
than two scaling variables fails to apply.
