  We discuss the solution theory of operators of the form $\nabla_X + A$,
acting on smooth sections of a vector bundle with connection $\nabla$ over a
manifold $M$, where $X$ is a vector field having a critical point with positive
linearization at some point $p \in M$. As an operator on a suitable space of
smooth sections $\Gamma^\infty(U, \V)$, 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 $\nabla$, $X$ and $A$.
