= Solution
For $a=-1$, the <local minimum> at the origin has $f=1$. Its full <level set> is
$$
\boxed{y^2=\frac{x^4}{2}-x^2.}
$$
The connected component containing the minimum is just the isolated origin. The same level also has the outer branches $y=\pm\sqrt{x^4/2-x^2}$ for $|x|\geq\sqrt2$; these are different connected components, not curves approaching the origin.
Both <saddle points> $(\pm1,0)$ have $f=e^{1/4}$. Their common <level set> factors as
$$
y^2=\frac{x^4}{2}-x^2+\frac12=\frac{(x^2-1)^2}{2}.
$$
Its explicit equations are therefore
$$
\boxed{y=\frac{x^2-1}{\sqrt2}\quad\text{or}\quad y=-\frac{x^2-1}{\sqrt2}.}
$$
The two <parabolas> cross at each saddle, explaining the change of contour connectivity there. Each displayed equation can be restricted to the square of part (ii) if only that sketch window is wanted.
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