  We describe work on solutions of certain non-divergence type and therefore
non-variational elliptic and parabolic systems on manifolds. These systems
include Hermitian and affine harmonics which should become useful tools for
studying Hermitian and affine manifolds, resp. A key point is that in addition
to the standard condition of nonpositive image curvature that is well known and
understood in the theory of ordinary harmonic maps (which arise from a
variational problem), here we also need in addition a global topological
condition to guarantee the existence of solutions.
