  Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1},
\dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ and
$m\in\mathbb{R}.$ We study the center-focus problem on its origin which is a
monodromic nilpotent critical point. By directly calculating the generalized
Lyapunov constants, we find that the origin is always a focus and we complete
the classification of its stability. This includes the most difficult case:
$s=kl$ and $m=(2k+1)!!/(2kl+1)!_{(2l)}.$ In this case, we prove that the origin
is always unstable. Our result extends and completes a previous one.
