An estimate from above of the number of invariant straight lines of $n$-th degree polynomial vector field

It is shown that the $n$-th degree polynomial vector field in the plane has at most $2n + 1$ ($2n + 2$) invariant straight lines when $n$ is even (odd) and $n\geq 3$ if it has a singular point for which $n + 1$ invariant straight lines and $n$ parallel invariant straight lines with a certain angular coefficient are incident.