= Solution
The completed <contour plot> is shown in the right-hand panel. Its geometry follows from the symmetries $f(-x,-y)=f(x,y)$ and $f(y,x)=f(x,y)$, together with $f\to0$ as $x^2+y^2\to\infty$. For orientation, the two <local minima> have value $-2e^{-3/2}$, the two <local maxima> have value $2e^{-1/2}$, and the <saddle point> has value one.
Negative <level sets> between $-2e^{-3/2}$ and zero form separate closed curves around the two <local minima> in the first and third quadrants. Positive <level sets> below one form a single closed curve enclosing the origin and both <local maxima>. At level one, a critical <level set> crosses itself at the origin and surrounds the two <local maxima> in two lobes. Between one and $2e^{-1/2}$, the <level sets> split into separate closed curves around those <local maxima>. The zero <level set> remains the two unbounded branches $xy=1/4$.
Near the origin, the <Taylor expansion> is $f=1-x^2-4xy-y^2+O((x^2+y^2)^2)$. Consequently the crossing directions at the saddle height satisfy $y=(-2\pm\sqrt3)x$, giving the local shape of the critical <level set>.
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