Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces

We show that, for all $\alpha\geq 0$, the generalized Grushin plane $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[\alpha ]+2}$, where $[\alpha]$ is the integer part of $\alpha$. The target dimension is sharp. This generalizes a recent result of Wu.