X-Elliptic harmonic maps

We study $X$-elliptic harmonic maps of an open set $\mathcal{U} \subset \mathbb{R}^n$ endowed with a family of vector fields $X =\ { X_1, ⋯ , X_m \}$ into a Riemannian manifold $S$ i.e. $C^\infty$ solutions $\phi: \mathcal{U} \to S$ to the nonlinear system $-L \phi^\alpha + a^{ij}(\Gamma^\alpha_{\beta \gamma} \circ \phi) (\partial_{x^i} \phi^\beta )(\partial_{x^j} \phi^\gamma) = 0$ where $L= \sum_{i,j=1}^N \partial_{x_j} (a^{ij} (x) \partial_{x_j} u)$ is an uniformly $X$-elliptic operator. We establish a Solomon type (cf. Solomon, J Differ Geom 21:151–162, 1985) result for $X$-elliptic harmonic maps $\phi: \mathcal{U} \to S^M \setminus \Sigma$ with values into a sphere and omitting a codimension two totally geodesic submanifold $\Sigma \subset S^M$. As an application of Harnack inequality (for positive solutions to $Lu = 0$) in Gutiérrez and Lanconelli (Commun Partial Differ Equ 28:1833–1862, 2003) we prove openness of $X$-elliptic harmonic morphisms.