= Solution
The critical-point equations are $2xy=0$ and $x^2+y^2=1$, so the points are $(\pm1,0)$ and $(0,\pm1)$. The Hessian is
$$
\begin{pmatrix}2y&2x\\2x&2y\end{pmatrix}.
$$
Thus $(0,1)$ is a strict minimum, $(0,-1)$ a strict maximum, and $(\pm1,0)$ are saddles. The local contour patterns are respectively ellipses, inverted ellipses, and crossing hyperbolas.
The <gradient flow> has $\dot x=-2xy$ and $\dot y=1-x^2-y^2$. Hence
$$
2xy\frac{dy}{dx}-y^2=x^2-1.
$$
Putting $u=y^2$ gives $xu'-u=x^2-1$. Multiplication by the <integrating factor> $1/x$ yields
$$
\left(\frac ux\right)'=1-\frac1{x^2},
$$
and therefore the general trajectory is
$$
\boxed{y^2=x^2+1+cx.}
$$
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