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We discuss the solution theory of operators of the form del(x) + A, acting on smooth sections of a vector bundle with connection del over a manifold M, where X is a vector field having a critical point with positive linearization at some point p is an element of M. As an operator on a suitable space of smooth sections Gamma(infinity)(U, nu), 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 del, X and A.