Quartic double well with bilinear coupling ($g(x,y)=x^4+y^4-4a^{2}xy,a>0$)

The stationary points of this polynomial are the origin and $(a,a)$, $(-a,-a)$. Its Hessian matrix at the origin has eigenvalues $\pm 4a^2$, so the origin is a saddle point. At each of the other points the eigenvalues are $8a^2$ and $16a^2$, giving strict local minima. The identityshows that they are also the only global minima, with value $-2a^4$.

For $-2a^4<c<0$, the level set $g=c$ has two closed components, one surrounding each minimum. The zero level set has two lobes meeting at the origin with the coordinate axes as tangent directions. For $c>0$, there is one closed outer component. These statements also follow in polar coordinates from the quadratic equation in $r^2$, together with the absence of other stationary points.