Meromorphic vector fields with single-valued solutions on complex surfaces

We study ordinary differential equations in the complex domain given by meromorphic vector fields on K"ahler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in particular, if it has an entire one) then, up to a bimeromorphic transformation, either the vector field is holomorphic or it preserves a fibration.