= Solution
The <Hessian matrix> is
$$
H(x,y)=\begin{pmatrix}1+2y&2x\\2x&1-2y\end{pmatrix}.
$$
At the origin it is the identity, so the origin is a strict \b[<local minimum>]. At $(0,1)$ it is $\operatorname{diag}(3,-1)$, so this point is a \b[<saddle point>]. At either remaining <stationary point>,
$$
H=\begin{pmatrix}0&\pm\sqrt3\\\pm\sqrt3&2\end{pmatrix},\qquad \det H=-3<0,
$$
which again gives a \b[<saddle point>]. All three <saddle points> have <eigenvalues> $3$ and $-1$; there are no <local maxima>.
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