Vector fields with a non-degenerate source

We discuss the solution theory of operators of the form ∇X+A∇X+A, acting on smooth sections of a vector bundle with connection ∇∇ over a manifold MM, where XX is a vector field having a critical point with positive linearization at some point p∈Mp∈M. As an operator on a suitable space of smooth sections Γ∞(U,V)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 ∇∇, XX and AA.