Integrability and Linearizability of Three Dimensional Vector Fields

Abstract

This paper deals with the integrability and linearizability problem of three dimensional systems $$\begin{aligned} \dot{x}&= x(1 +ax+by+cz),\\ \dot{y}&= -y+ dx^2 +exy + fxz+gyz+hy^2+kz^2,\\ \dot{z}&= z(1 + \ell x + my +pz). \end{aligned}$$ More precisely, we give a complete set of necessary conditions for integrability and linearizability and then prove their sufficiency using Darboux method and Darboux inverse Jacobi multiplier, power series argument and a solution of a Riccati equation.