Continuation of integral curves of second-order ordinary differential equations to singular sets via a local first integral (Q2455129)

Consider the differential system in the plane \[ a(x,y)\dot y= P(x,y),\quad \dot x= y,\tag{\(*\)} \] where \(a\) and \(P\) are continuously differentiable in \(G\subset R^2\). \[ S_1:= \{(x,y)\in G: a(x,y)= 0,\,P(x,y)\neq 0\} \] is called the singualr set of first type of \((*)\), \[ S_2:= \{(x,y)\in G: P(x,y)= 0,\, ya(x,y)\neq 0\} \] is called the singular set of second type of \((*)\). The authors investigate the continuation of integral curves of \((*)\) to points of \(S_1\) and \(S_2\).