In this paper, we are concerned with a critical Grushin type equation in \(\mathbb {R}^{K+N}\) : $\mathfrak {p}$ \(\begin{aligned} -\Delta u(x)=Q(x)\frac{u^{2^\star -1}(x)}{|y|},\ u>0,\ x=(y,z)\in \mathbb {R}^K\times \mathbb {R}^{N}, \end{aligned}\) where Q is a nonnegative and bounded function with a stable critical point, and \(2^\star :=\frac{2(K+N-1)}{K+N-2}\) . We first prove the existence of infinitely many non-radial positive solutions of \((\mathfrak {p})\) . The bubbles of these solutions are located near a cylindrical surface and symmetric with respect to the plane \(z_6=0\) . Next, we prove that these positive solutions are non-degenerate. Precisely, the linearized equation of \((\mathfrak {p})\) has only a zero solution in a suitable space. Last, we glue together bubbles with different concentration rates to obtain new multi-bubbling solutions of \((\mathfrak {p})\) . The methods we used mainly are Lyapunov-Schmidt reduction and local Pohozaev identities.
