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Since $Y={\int }_0^{\infty }{1}_{\{Y\ge y\}}dy$, Tonelli's theorem givesFor the integer-valued minimum,Markov's inequality now gives

The critical-point equations are $2xy=0$ and $x^2+y^2=1$, so the points are $(\pm 1,0)$ and $(0,\pm 1)$. The Hessian isThus $(0,1)$ is a strict minimum, $(0,-1)$ a strict maximum, and $(\pm 1,0)$ are saddles. The local contour patterns are respectively ellipses, inverted ellipses, and crossing hyperbolas.

The gradient flow has $\dot{x}=-2xy$ and $\dot{y}=1-x^2-y^2$. HencePutting $u=y^2$ gives $x{u}^{\prime }-u=x^2-1$. Multiplication by the integrating factor $1/x$ yieldsand therefore the general trajectory is