Nowadays it is common to see PDEs defined as either closed embedded submanifolds of a jet bundle (of some appropriate order) or else, if some further conditions are satisfied, as a fibered submanifold. In some earlier literature, though (e.g., Quillen's dissertation), a PDE is defined as a subbundle of a jet bundle. That strikes me as a strong restriction, but I don't immediately see how strong. I presume it is related to whether solutions exist.
Is anyone aware of good resources discussing when a PDE is a subbundle, rather than a fibered submanifold? Are there easy-to-check conditions on the equation defining the submanifold that ensure this?
