Bifurcations of certain family of planar vector fields tangent to axes (Q1105108)

This interesting paper deals with the families of planar vector fields of the following form: \(V_{\mu}(x,y)=[xW_ 1(x,y;\mu),yW_ 2(x,y;\mu)],\) where \(\mu =(\mu_ 1,\mu_ 2)\in {\mathbb{R}}^ 2\). If \(dV_ 0(0)=0\), the vector field \(V_ 0\) is called singular. The author proves that the space of germs of deformations of singular vector fields is divided into degenerate germs and nondegenerate germs such that the degenerate germs form a union of a finite number of submanifolds of codimension one. Every nondegenerate deformation is \(C^ 0\)-orbitally equivalent to the family \[ M_{\mu}(x,y)=[x(\mu_ 1+x+y),y(\mu_ 2-(\alpha +1)x/\beta -\alpha y/(\beta +1)+\nu R(x,y)], \] where R(x,y) is a specially chosen perturbation. The bifurcation diagram of the family \(M_{\mu}(x,y)\) is investigated in detail. Some new resuls about the function \(h\mapsto \iint_{H\leq h}Pdxdy\), where P(x,y) and H(x,y) are not polynomials, are obtained.