  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.
