The leading term of the first return map of a singular point with a fixed Newton diagram (Q1365711)

Analytic vector fields of dynamic systems \[ \begin{aligned} y\dot x & = X(x,y)= \sum a_{ij}x^i y^j,\\ x\dot y & = Y(x,y)= \sum b_{ij}x^i y^j,\quad (a_{ij}, b_{ij})\neq(0,0),\end{aligned} \] are considered. The coefficient of the leading term in the asymptotics of the Poincaré function of the vector fields with a fixed Newton diagram \(\Gamma\) is computed. The \(\Gamma\)-nondegenerate vector fields satisfying two conditions of nondegeneracy are investigated.