Subelliptic harmonic morphisms

We study subelliptic harmonic morphisms i.e. smooth maps $\phi: \Omega \to \tilde\Omega$ among domains $\Omega \subset \mathbb{R}^n$ and $\tilde\Omega \subset \mathbb{R}^M$ endowed with Hörmander systems of vector fields $X$ and $Y$, that pull back local solutions to $H_Y v = 0$ into local solutions to $H_X u = 0$, where $H_X$ and $H_Y$ are Hörmander operators. We show that any subelliptic harmonic morphism is an open mapping. Using a subelliptic version of the Fuglede-Ishihara theorem (due to E. Barletta [5]) we show that given a strictly pseudoconvex CR manifold $M$ and a Riemannian manifold $N$ for any heat equation morphism $\Psi: M \times (0, \infty) \to N \times (0, \infty)$ of the form $\Psi(x,t) = ( \phi (x), h(t))$ the map $\phi : M \to N$ is a subelliptic harmonic morphism.