  In this paper, we establish Liouville theorems for the following system of
elliptic differential inequalities $$ \Delta_{\mathbb H}u^{m_1}+|\eta|_{\mathbb
H}^{\gamma_1}|v|^p\leq0,$$ $$ \Delta_{\mathbb H}v^{m_2}+|\eta|_{\mathbb
H}^{\gamma_2}|u|^q\leq0,$$ on different unbounded open domains of Heisenberg
group $\mathbb H$, including the whole space, and half space of $\mathbb H$.
Here $p>m_2>0$, $q>m_1>0$.
