Connections and genuinely ramified maps of curves

Given a singular connection D on a vector bundle E over an irreducible smooth projective curve X, defined over an algebraically closed field, we show that there is a unique maximal subsheaf of E on which D induces a nonsingular connection. Given a generically smooth map phi : Y -> X between irreducible smooth projective curves, and a singular connection (V, D) on Y, the direct image phi(*) V has a singular connection. Let R(phi (*) O-Y) be the unique maximal subsheaf on which the singular connection on phi(*) O-Y - corresponding to the trivial connection on O-Y - induces a nonsingular connection. We prove that the homomorphism of etale fundamental groups phi(*) : pi(et)(1)(Y, y(0))-, pi(et)(1) (X, phi(y(0))) induced by phi is surjective if and only if O-X subset of R(phi(*) O-Y) is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism phi(*) is surjective if and only if O-X = R((phi(*) O-Y). For any nonsingular connection D on a vector bundle V over X, there is a natural map V -> R(phi(*) phi*V). When the characteristic of the base field is zero, we prove that the map ? is genuinely ramified if and only if V = R(phi(*) phi*V).