On integrability conditions and limit cycle bifurcations for polynomial systems

We present an approach to finding general conditions for integrability of a given family of two-dimensional polynomial systems using conditions computed when some parameters were fixed. We apply it to obtain integrability conditions for a Lotka-Volterra planar complex quartic system having homogeneous nonlinearities. We also study bifurcations of limit cycles from each component of the center variety of the corresponding quartic real system.