Given vector fields $Y$, $Z$ on a (possibly compact) manifold $M$, I would like to know about the existence of solutions $X$ to the differential equation $$ \nabla_Y X + a \cdot \mathrm{div}(Y)\cdot X + b X = Z$$ where $\nabla$ denotes the Levi-Civita connection.

What about local and/or global solutions? What if $Y = \mathrm{grad} \phi$ for some function $\phi$?

In the case that Y is the radial vector field around some special point, this is easily transferred into a normal ODE. But I don't really see what to do in a more general situation.

*\Edit: So, as I should have seen myself, existence of a local solution is quite clear because this is just a system of ODEs along the integral curves of V. Furthermore, you can say basically nothing about global solutions. So, let us concentrate on local solutions.*

**Another question though:**

It turns out, I have to solve the equation locally, but around the points where $Y$ vanishes. And now it doesn't seem clear at all, what happens.

Because, if $Y$ behaves like the radial vector field, then you basically have to solve singular ODEs, but if $Y$ rotates around $p$ (like $- \mathrm{sin}(x^1) \partial_1 + \mathrm{cos}(x^2) \partial_2$ around $0$), then it is not clear at all to me how I can ensure that the solutions fit together smoothly at $p$.