A physical system on a flat configuration space with holonomic constraints can be reformulated as a system on a curved configuration space with fewer dimensions and no constraints.
By the Whitney embedding theorem, the converse holds: unconstrainted physical systems on curved configuration spaces can be reformulated as systems on flat configuration spaces of higher dimension with holonomic constraints.
In both cases, the phase space of the physical system is the cotangent bundle of the configuration space.
However, not all phase spaces are cotangent bundles, e.g., the phase space of a spinning top with fixed spin is the 2-sphere, which is compact and therefore necessarily not a cotangent bundle.
My question: Can a physical systems whose curved phase space is not a cotangent bundle always be reformulated as a system on a flat configuration space with non-holonomic constraints, and vice versa? In other words, if we equate a constrained system with its curved-space reformulation, are the properties "all constraints are holonomic" and "the phase space is a cotangent bundle" equivalent?