Darboux and rational first integrals for a family of cubic three dimensional system

Abstract EN

In this paper, we investigate the first integrals of the following system x = y, y = z, Z = b1x + b2 yx 2 + b3 z 3 where and , This kind of system is a special case of three-dimensional polynomial cubic differential systems.

Generally, several methods can be used to investigate the first integrals, but unfortunately, most of them are not enabled for finding first integrals.

In this study, the Darboux method has been used to study the first integrals for the generalized system for all parameters.

We characterize all its invariant algebraic surfaces and all its exponential factors of that system.

We have shown that the above system does not admit a polynomial, rational, and Darboux first integrals for any values of the parameters