How many are projectable classical linear connections with a prescribed Ricci tensor

How many are projectable classical linear connections with a prescribed Ricci tensor and trace of torsion on the total space fibered manifold? The questions answered in analytic case by using Cauchy-Kowalevski theorem. In C? case, we answer how 2-dimensional manifold. also deduce that any 2-form manifold at least fibres can be realized locally as connection.