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. © 1999 American Institute of Physics.
Cairó, L., Feix, M. R., & Llibre, J. (1999). Darboux method and search of invariants for the Lotka-Volterra and complex quadratic systems. Journal of Mathematical Physics, 40(4), 2074-2091. https://doi.org/10.1063/1.532852