Parabolic quartic critical-point classification

For $F(x,y)=(y-c{x}^2{)}^2+\alpha x^2-\beta x^4$ with $\beta >0$, its stationary points are the origin and, when $\alpha >0$, $x=\pm \sqrt{\alpha /(2\beta )}$, $y=c{x}^2$. The origin is a strict local minimum for $\alpha >0$ and a saddle point of a scalar function for $\alpha <0$. At $\alpha =0$ its Hessian matrix is singular, but the vertical line gives positive values and the parabola $y=c{x}^2$ negative values, so it remains a degenerate saddle. The other stationary points have Hessian determinant $-8\alpha$ and are saddles.