  Let $\Omega\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial
frame on $\Omega$, that is a set of $m$ linearly independent vector fields
prescribed on $\Omega$ ($m\leq n$). We consider the issue of describing the set
of all maps $F:\Omega\to\mathbb{R}^n$ with the property that each of the given
vector fields is an eigenvector of the Jacobian matrix of $F$. By introducing a
coordinate independent definition of the Jacobian, we obtain an intrinsic
formulation of the problem, which leads to an overdetermined PDE system, whose
compatibility conditions can be expressed in an intrinsic, coordinate
independent manner. To analyze this system we formulate and prove a
generalization of the classical Frobenius integrability theorems. The size and
structure of the solution set of this system depends on the properties of the
partial frame, in particular, whether or not it is in involution. A
particularly nice subclass of involutive partial frames, called rich, can be
completely analyzed. Involutive, but non-rich case is somewhat harder to
handle. We provide a complete answer in the case of $m=3$ and arbitrary $n$, as
well as some general results for arbitrary $m$. The non-involutive case is far
more challenging, and we only obtain a comprehensive analysis in the case
$n=3$, $m=2$. Finally, we provide explicit examples illustrating the various
possibilities. Our initial motivation for considering this problem comes from
the geometric study of hyperbolic conservative systems in one spatial
dimension.
