  We show how the Fourier transform for distributional sections of vector
bundles over symmetric spaces of non-compact type $G/K$ can be used for
questions of solvability of systems of invariant differential equations in
analogy to H\"ormander's proof of the Ehrenpreis-Malgrange theorem. We get
complete solvability for the hyperbolic plane $\mathbb{H}^2$ and partial
results for products $\mathbb{H}^2 \times \cdots \times \mathbb{H}^2$ and the
hyperbolic 3-space $\mathbb{H}^3$.
