  We study critical and stationary, i.e. critical with respect to both inner
and outer variations, points of polyconvex functionals of the form $f(X) =
g(\det(X))$, for $X \in \mathbb{R}^{2\times 2}$. In particular, we show that
critical points $u \in Lip(\Omega,\mathbb{R}^2)$ with $\det(Du) \neq 0$ a.e.
have locally constant determinant except in a relatively closed set of measure
zero, and that stationary points have constant determinant almost everywhere.
This is deduced from a more general result concerning solutions $u \in
Lip(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ to the linearized
problem $curl(\beta Du) = 0$. We also present some generalization of the
original result to higher dimensions and assuming further regularity on
solutions $u$. Finally, we show that the differential inclusion associated to
stationarity with respect to polyconvex energies as above is rigid.
