  We consider the vectorial system \[ \begin{cases} Du \in \mathcal{O}(2), &
\mbox{a.e. in}\;\Omega, u=0, & \mbox{on} \;\partial \Omega, \end{cases} \]
where $\Omega$ is a subset of $\R^2$, $u:\Omega\to \R^2$ and $\mathcal{O}(2)$
is the orthogonal group of $\R^2$. We provide a variational method to select,
among the infinitely many solutions, the ones that minimize an appropriate
weighted measure of the singular set of the gradient.
