Normal Form, Symmetry and Infinite Dimensional Lie Algebra for System of Ode's

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.