Differential equations with given partial and first integrals

© Springer International Publishing Switzerland 2016. In this chapter we present two different kind of results. First, under very general assumptions we characterize the ordinary differential equations in ℝN which have a given set of either M ≤ N, or M >N partial integrals, or M <N first integrals, or M ≤ N partial and first integrals. Second, in ℝN we provide some results on integrability, in the sense that the characterized differential equations admit N −1 independent first integrals.