On the solvability of planar complex linear vector fields

Abstract:

The article discusses the local solvability, or lack thereof, of vector fields whose coefficients are complex-valued linear functions in $\mathbb {R}^{2}$ (here called complex linear). It is proved that such a vector field is locally solvable in $\mathbb {R}^{2}$ if and only if it is locally solvable and does not have compact orbits in the complement of its critical set or, equivalently, its Meziani number is different from zero.