\begin{cases}<br />
x+y+uv=0 \\<br />
uxy+v=0<br />
\end{cases}

Are there any solutions to this system for for u, v in terms of x, y near the point (x,y,u,v)=(0)

We cannot apply the implicit function theorem here as the matrix of the partials is not invertible. Is it possible to do a change of coordinates so the jacobian matrix is invertible near (0,0,0,0)?