= Solution
Now $h=e^x-e^y-x+y$ and the origin has stationary value zero. Near it,
$$
h=\frac{x^2-y^2}{2}+\frac{x^3-y^3}{6}+O((|x|+|y|)^4).
$$
The <level curves> therefore resemble hyperbolas near the <saddle point of a scalar function>. Along the $x$-axis $h=e^x-x-1>0$ for $x\ne0$, while along the $y$-axis $h=1-e^y+y<0$ for $y\ne0$. The zero <level set> includes the exact diagonal $x=y$ and another branch crossing it at the origin; this second branch is tangent to $y=-x$ but is curved away from the origin. The right panel of the contour figure displays these features.
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