During the course of the evolution, we may however leave to domain of definition of the primary system, so that the whole set of equations need to be replaced by something else.

In a sense this is what I do but then in a local sense. At the singularity the equations are not defined so I construct regularized equations which are actually like a new set of equation which can be used in a neighbourhood of the singularity.

So I have something like a state space manifold where I have a special chart in the neighbourhood of the singularity.