  We show that the energy-momentum equations arising from inner variations
whose Lagrangian satisfies a generic symmetry condition are generically
ill-posed. This is done by proving that there exists a subclass of Lipschitz
solutions that are also solutions to a differential inclusion. In particular
these solutions can be nowhere C1. We prove that these solutions are not
stationary points if the Lagrangian W is C1 and strictly rank-one convex. In
view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for
solution of the energy-momentum equation in dimension 2 we give a sufficient
condition for the non-existence of a partial C1-regularity result even under
the condition that the mappings satisfy a positive Jacobian determinant
condition. Finally we consider a number of well-known functionals studied in
nonlinear elasticity and geometric function theory and show that these do not
satisfy this obstruction to partial regularity.
