  We consider the following elliptic system
  \Delta u =\nabla H (u) \ \ \text{in}\ \ \mathbf{R}^N, where
$u:\mathbf{R}^N\to \mathbf{R}^m$ and $H\in C^2(\mathbf{R}^m)$, and prove, under
various conditions on the nonlinearity $H$ that, at least in low dimensions, a
solution $u=(u_i)_{i=1}^m$ is necessarily one-dimensional whenever each one of
its components $u_i$ is monotone in one direction. Just like in the proofs of
the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in
dimension 3 (Ambrosio-Cabr\'{e}), the key step is a Liouville theorem for
linear systems. We also give an extension of a geometric Poincar\'{e}
inequality to systems and use it to establish De Giorgi type results for stable
solutions as well as additional rigidity properties stating that the gradients
of the various components of the solutions must be parallel. We introduce and
exploit the concept of {\it an orientable system}, which seems to be key for
dealing with systems of three or more equations. For such systems, the notion
of a stable solution in a variational sense coincide with the pointwise (or
spectral) concept of stability.
