Summary: In this paper, we study a system of the form \[ \begin{cases} - \Delta_{\lambda} u = v \\ - \Delta_{\lambda} v = u^p \end{cases} \text{ in } \mathbb{R}^N, \] where \(p \in \mathbb{R}\), and \(\Delta_{\lambda}\) is a sub-elliptic operator defined by \[ \Delta_{\lambda} = \sum\limits_{i = 1}^N \partial_{x_i} \left(\lambda_i^2 \partial_{x_i}\right). \] Under some general hypotheses of the functions \(\lambda_i, i = 1,2, \ldots, N\), we first prove that the system has no positive super-solution when \(p \leq 1\). In the case \(p > 1\), we establish a Liouville type theorem for the class of stable positive solutions. This result is an extension of some result in [H. Hajlaoui et al., Pac. J. Math. 270, No. 1, 79–93 (2014; Zbl 1301.35051)] for the case of Laplace operator.