Existence of solutions for system of first order linear inhomogeneous PDEs

See my question on MO.

For $i=1,\dots ,r$, let $Z_i$ be $r$ linearly independent vector fields defined on an open subset $U$ of ${\mathbb{R}}^n$, $r<n$. And let ${\lambda }_i$ be $r$ smooth functions defined on $U$. We can consider the inhomogeneous system of first-order linear PDEs

Question: What hypothesis do we need to assure the local existence of a solution $f$?

According to Bryant answer (remark at the end): it can be shown the existence using the involutivity of $\{Z_i\}$, by using Frobenius theorem.

But also it can be proven in general, not only for linear inhomogeneous PDEs, but for first order PDEs whenever they give rise to a "involutive submanifold" $\mathrm{\Sigma }$ of the first order jet bundle. I don't know yet what is an involutive manifold.