Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

This paper study the type of integrability differential systems with separable variables $\dot{x}=h\left (x\right )f\left (y\right )$ , $\dot{y}= g\left where $h$ $f$ and $g$ are polynomials. We provide a criterion for existence generalized analytic first integrals such systems. Moreover we characterize polynomial all In particular case $h\left )=\left (ax+b\right )^{m}$ necessary sufficient conditions in order that this subclass has integral. These results extend known from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) Llibre Valls Syst., Ser. B 20:2657–2661, 2015). Such important due to fact after blow-up change any planar quasi-homogeneous system can be transformed into special $\dot{x}=xf\left $\dot{y}=g\left