Integrability and zero-Hopf bifurcation in the Sprott A system

The first objective of this paper is to study the Darboux integrability of the polynomial differential system x˙=y,y˙=−x−yz,z˙=y²−a and the second one is to show that for a>0 sufficiently small this model exhibits one small amplitude periodic solution that bifurcates from the origin of coordinates when a=0. This model was introduced by Hoover as the first example of a differential equation with a hidden attractor and it was used by Sprott to illustrate a differential equation having a chaotic behavior without equilibrium points, and now this system is known as the Sprott A system.