On the Integrability of Persistent Quadratic Three-Dimensional Systems

We consider a nine-parameter familiy of 3D quadratic systems, x˙=x+P2(x,y,z), y˙=−y+Q2(x,y,z), z˙=−z+R2(x,y,z), where P2,Q2,R2 are quadratic polynomials, in terms of integrability. We find necessary and sufficient conditions for the existence of two independent first integrals of corresponding semi-persistent, weakly persistent, and persistent systems. Unlike some of the earlier works, which primarily focus on planar systems, our research covers three-dimensional spaces, offering new insights into the complex dynamics that are not typically apparent in lower dimensions.