This is the research project of a PhD student of mine. Actually the general project is sufficiently large for a number of PhD theses. There are lots of open questions in the non-Hamiltonian case.
What my student is doing is to study simplest (mostly low-dimensional) cases, and with only nondegenerate singularities:
– Systems of type (1,1), i.e. 1 vector field and 1 function, dimension 2. Already in this case, the picture is non-trivial.
– Any dimension: a real classification of nondegenerate singular points of type (n,0), i.e. zero function and n vector fields
– Dimension 3: local strcture of type (1,2), type (2,1), and also some questions about the global structure.
– Dimension 4: Monodromy phenomenon around certain singularities in dimension 4.
That will be enough for his thesis. He must write up a research article before summer.