Stationary points for surface

Find the stationary points for the surface. And find the local and absolute maxima or minima for the following function.

$\displaystyle f(x,y)= xye^{-1/2(x^2+y^2)}$

Is there any stationary points for this surface?

Let z = f(x,y)

I found the

$\displaystyle \delta{z}/\delta{x}$ = $\displaystyle y[e^{-1/2(x^2+y^2)} - x^2e^{-1/2(x^2+y^2)}]$

and

$\displaystyle \delta{z}/\delta{x}$ = $\displaystyle x[e^{-1/2(x^2+y^2)} - y^2e^{-1/2(x^2+y^2)}]$

I'm supposed to equate the two equations to get the stationary points (X , Y) and substitute them into f(x,y) to get Z. But I could not since I could only get $\displaystyle y - x = y^2 - x^2$ from the two equations and that's it.

How do I proceed? Did i do anything wrong?