Algorithmic reduction of Poincare-Dulac normal forms and Lie algebraic structure

The Poincare-Dulac normal form of a given resonant system is in general non unique; given a specific normal form, one would like to further reduce it t o a simplest normal form. In this Letter we give an algorithm, based on the Lie algebraic structure of the set of normal forms, to obtain this. The al gorithm can be applied under some condition, nongeneric but often met in ap plications. When applicable, it is only necessary to solve linear equations , and is more powerful than the one proposed in previous work by the same a uthor [Lett. Math. Phys. 42 (1999), 103-114; and Ann. Inst. H. Poincare Phy s. Theor. 70 (1999), 461-514].