$\displaystyle \begin{cases}

x+y+uv=0 \\

uxy+v=0

\end{cases}$

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

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)?