= Solution
The <Hessian matrix> is
$$
H=\begin{pmatrix}30x^4-48y^2&-96xy\\-96xy&30y^4-48x^2\end{pmatrix}.
$$
At each nonzero <stationary point>, its diagonal entries are $1536$ and the off-diagonal entries are $\pm768$. Its <eigenvalues> are $768$ and $2304$, both positive. Thus the <Hessian matrix> is <positive-definite matrix>[positive definite], and the <second-derivative test> proves that
$$
\boxed{\text{all four nonzero stationary points are strict local minima}}.
$$
Their common value is $h=-512$. In fact they are <global minima>: with $u=|xy|$, the <arithmetic-geometric mean inequality> gives $x^6+y^6\geq2u^3$, so
$$
h\geq2u^3-24u^2\geq-512.
$$
The last one-variable <polynomial> attains its minimum at $u=8$, and equality in the first bound requires $|x|=|y|$, precisely the four points above.
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