An inverse approach to the center-foci problem

The classical Center-Focus Problem posed by H. Poincar'e in 1880’s is concerned on the characterization of planar polynomial vector fields $X=(-y+P(x,y))\dfrac{\partial}{\partial x}+(x+Q(x,y))\dfrac{\partial}{\partial y},$with$P(0,0)=Q(0,0)=0,$ such that all their integral trajectories are closed curves whose interiors contain a fixed point called center or such that all their integral trajectories are spirals called foci. In this paper we state and study the inverse problem to the Center-Foci Problem i.e., we require to determine the analytic planar vector fields $X$ in such a way that for a given Liapunov function $$V=V(x,y)={\displaystyle \frac{\lambda }{2}}(x^2+y^2)+{\displaystyle \sum _{j=3}^{\mathrm{\infty }}{H}_j(x,y),}$$ where $H_j=H_j(x,y)$ are homogenous polynomial of degree $j,$ the following equation holds $$X(V)={\displaystyle \sum _{j=3}^{\mathrm{\infty }}{V}_j(x^2+y^2{)}^{j+1},}$$ where $V_j$ for $j\in \mathbb{N}$ are the Liapunov constants. In particular we study the case when the origin is a center and the vector field is polynomial.