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 $$V=V(x,y)=\dfrac{\lambda}{2}(x^2+y^2)+\displaystyle\sum_{j=3}^{\infty} H_j(x,y),$$ where $H_j=H_j(x,y)$ are homogenous polynomial of degree $j,$ the following equation holds $$X(V)=\displaystyle\sum_{j=3}^{\infty}V_j(x^2+y^2)^{j+1},$$ 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.