  Pseudo-Riemannian metrics with Levi-Civita connection in the projective class
of a given torsion free affine connection can be obtained from (and are
equivalent to) the maximal rank solutions of a certain overdetermined
projectively invariant differential equation often called the metrizability
equation. Dropping this rank assumption we study the solutions to this equation
given less restrictive generic conditions on its prolonged system. In this
setting we find that the solution stratifies the manifold according to the
strict signature (pointwise) of the solution and does this in way that locally
generalizes the stratification of a model, where the model is, in each case, a
corresponding Lie group orbit decomposition of the sphere. Thus the solutions
give curved generalizations of such embedded orbit structures. We describe the
smooth nature of the strata and determine the geometries of each of the
different strata types; this includes a metric on the open strata that becomes
singular at the strata boundary, with the latter a type of projective infinity
for the given metric. The approach reveals and exploits interesting highly
non-linear relationships between different linear geometric partial
differential equations. Apart from their direct significance, the results show
that, for the metrizability equation, strong results arising for so-called
normal BGG solutions, and the corresponding projective holonomy reduction,
extend to a far wider class of solutions. The work also provides new results
for the projective compactification of scalar-flat metrics.
