Viewed on a different time scale, i.e. under the (deterministic) change of time given by $\tau(t) = (t - 2 \alpha t )^{\frac{1}{1-2 \alpha}}$ (so that $\tau'(t) = \tau(t)^{2 \alpha}$), the SDE simplifies to (in a distributional sense): $$ d \tilde{X}_t = b(\tilde{X}_t) d B_t $$ where $\tilde{X}_t = X_{\tau(t)}$ and $B_t$ is a standard Brownian motion. This time change makes sense for $\alpha \in [0, 1/2)$ and $t>0$. Sufficient conditions for existence and uniqueness of the time-changed process can be found in, e.g., Section 1.2 of the monograph
Cherny, Alexander S.; Engelbert, Hans-Jürgen, Singular stochastic differential equations., Lecture Notes in Mathematics 1858. Berlin: Springer (ISBN 3-540-24007-1/pbk). viii, 128 p. (2005). ZBL1071.60003.