Meromorphic vector fields with single-valued solutions on complex surfaces

Abstract: 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.