Maximum number of limit cycles of the equation \((b_{20}x^2 + b_{11} xy + b_{02}y^2 + b_{00} ) dy = (a_{20}x^2 + a_{11} xy + a_{02}y^2 + a_{00} ) dx\) is two (Q2187854)

Consider the polynomial system \begin{align*} \begin{split} \frac{dx}{dt} &=a_{20}x^2 + a_{11}xy + a_{02}y^2 +a_{00},\\ \frac{dx}{dt} &=b_{20}x^2 + b_{11}xy + b_{02}y^2 +b_{00}. \end{split}\tag{1} \end{align*} The author proves that, for any real parameters \(a_{00}, b_{00}, a_{ij}, b_{ij}, i+j=2\), the maximum number of limit cycles of system (1) is two. The proof consists in completing results by the author [Differ. Equ. 50, No. 12, 1685--1687 (2014; Zbl 1317.34041); translation from Differ. Uravn. 50, No. 12, 1680--1682 (2014)] and \textit{L. A. Cherkas} [Differ. Equations 9, 1099--1103 (1975; Zbl 0316.34027); translation from Differ. Uravn 9, 1432--1437 (1973)].