Darboux method and search of invariants for the Lotka–Volterra and complex quadratic systems

The Darboux method introduces algebraic solutions quite useful to obtain invariant and first integrals of polynomial differential systems. Here we study the 2D Lotka–Volterra (LVS) and the complex quadratic system (QS) using straight lines for both and conics for the LVS. The conditions needed to obtain these invariants are given and a study of the phase space portrait is done.