We discuss the solution theory of operators of the form ∇_X + A, acting on smooth sections of a vector bundle with connection ∇ over a manifold M, where X is a vector field having a critical point with positive linearization at some point p ∈ M. As an operator on a suitable space of smooth sections Γ^∞(U, V), it fulfills a Fredholm alternative, and the same is true for the adjoint operator. Furthermore, we show that the solutions depend smoothly on the data ∇, X and A.