This paper study the type of integrability of differential systems with separable variables \(\dot{x}=h\left (x\right )f\left (y\right )\), \(\dot{y}= g\left (y\right )\), where \(h\), \(f\) and \(g\) are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.

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

本文研究了具有可变量\（\ dot {x} = h \ left（x \ right）f \ left（y \ right）\），\（\ dot {y} = g \ left（y \ right）\），其中\（h \），\（f \）和\（g \）是多项式。我们为此类微分系统的广义解析第一积分的存在提供了一个标准。此外，我们表征了所有此类系统的多项式可积性。

在特殊情况下，（\ h \ left（x \ right）= \ left（ax + b \ right）^ {m} \）我们提供必要和充分的条件，以使该系统子类具有广义的解析第一积分。这些结果扩展了Giné等人的已知结果。（Discrete Contin。Dyn。Syst。33：4531–4547，2013）和Llibre和Valls（Discrete Contin。Dyn。Syst。，Ser。B 20：2657–2661，2015）。这种可分离变量的微分系统之所以重要，是因为以下事实：在变量发生爆炸变化之后，任何平面准齐次多项式微分系统都可以转换为可分离变量的特殊微分系统\（\ dot {x} = xf \左（y \ right）\），\（\ dot {y} = g \左（y \ right）\），带有\（f \）和\（g \） 多项式。