In this paper using theory of linear operators and normal forms we generalize a result of Poincaré about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta  about local first integrals of semi-quasi-homogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of their first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.
- Differential system
- Local first integral
- Normal form
Li, W., Llibre, J., & Zhang, X. (2003). Local first integrals of differential systems and diffeomorphisms. Zeitschrift fur Angewandte Mathematik und Physik, 54, 235-255. https://doi.org/10.1007/s000330300003