  In this work the following energy is considered
$I(u)=\int\limits_B{\frac{1}{2}|\nabla u|^2+\rho(\det\nabla u)\;dx},$ where
$B\subset\mathbb{R}^2$ denotes the unit ball, $u\in W^{1,2}(B,\mathbb{R}^2),$
and $\rho:\mathbb{R}\rightarrow\mathbb{R}_0^+$ smooth and convex with
$\rho(s)=0$ for all $s\le0$ and $\rho$ becomes affine when $s$ exceeds some
value $s_0>0.$ Additionally, we may impose $(M\in
\mathbb{N}\setminus\{0\})-$covering maps as boundary conditions in a suitable
fashion. For such situations we then construct radially symmetric $M-$covering
stationary points of the energy, which are at least $C^1$ (in some
circumstances even $C^\infty),$ and verify more refined properties, which these
stationary points need to satisfy. We do so by following the strategy first and
foremost developed by P. Bauman, N. C. Owen, and D. Phillips (BOP) confirming
and generalising that the method remains valid beyond the $M=2-$case for an
arbitrary $M.$ Furthermore, as far as we know, this is the first treatise of
BOP-theory in finite elasticity. The finiteness, not imposing such strict
conditions, allows for a richer class of possible behaviours of the stationary
points, making it more difficult to completely determine them.
