  We investigate here the following weighted degenerate elliptic system
\begin{align*} -\Delta_{s} u =
\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}} v^p, \quad
-\Delta_{s} v =
\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}}u^\theta, \quad
u,v>0\quad\mbox{in }\; \mathbb{R}^N:=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.
\end{align*} where $\Delta_{s}=\Delta_{x}+|x|^{2s}\Delta_{y},$ is the Grushin
operator, $s \geq 0,$ $\alpha \geq 0$ and $1<p\leq\theta.$ Here
$$\|\mathbf{x}\|=\Big(|x|^{2(s+1)}+|y|^2\Big)^{\frac{1}{2(s+1)}},
\;\mbox{and}\;\; \mathbf{x}:=(x, y)\in \mathbb{R}^N:=\mathbb{R}^{N_1}\times
\mathbb{R}^{N_2}.$$ In particular, we establish some new Liouville-type
theorems for stable solutions of the system, which recover and considerably
improve upon the known results \cite{cow, Hfh, HU, Fa, DP}. As a consequence,
we obtain a nonexistence result for the weighted Grushin equation
\begin{align*} -\Delta_{s} u
=\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}} u^p,\;\; \quad u>0
\quad \mbox{in }\;\; \mathbb{R}^N. \end{align*}
