Weak total rigidity for polynomial vector fields of arbitrary degree

We prove that in the space of the polynomial vector fields of arbitrary degree $n$ with $n+1$ different singular points at infinity the set of vector fields that are orbitally topologically equivelent to a generic vector field (modulo affine equivalence) is no more than countable.
This is the second one of two closely related papers. It was started after the first one, “Total rigidity of generic quadratic vector fields”, was completed. The present paper is motivated by the problem stated at the end of the first paper. The problem remains open. A slightly weaker problem is solved below.
This paper is independent on the first one. For the sake of convinience, it is published first.