  We study the symmetry and uniqueness of maps which minimise the
$np$-Dirichlet energy, under the constraint that their Jacobian is a given
radially symmetric function $f$. We find a condition on $f$ which ensures that
the minimisers are symmetric and unique. In the absence of this condition we
construct an explicit $f$ for which there are uncountably many distinct energy
minimisers, none of which is symmetric. Even if we prescribe the maps to be the
identity on the boundary of a ball we show that the minimisers need not be
symmetric. This gives a negative answer to a question of H\'{e}lein (Ann. Inst.
H. Poincar\'{e} Anal. Non Lin\'{e}aire 11 (1994), no. 3, 275-296).
