Stability for a family of planar systems with nilpotent critical points

Consider a family of planar polynomial systems x = y²ˡ⁻¹ - x²ᵏ⁺¹, y =-x +m y²ˢ⁺¹, where l,k,s^*, 2≤ l ≤ 2s and m. 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.