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