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 

<cite authors="Cherny, Alexander S.; Engelbert, Hans-Jürgen">_Cherny, Alexander S.; Engelbert, Hans-Jürgen_, [**Singular stochastic differential equations.**](http://dx.doi.org/10.1007/b104187), Lecture Notes in Mathematics 1858. Berlin: Springer (ISBN 3-540-24007-1/pbk). viii, 128&nbsp;p. (2005). [ZBL1071.60003](https://zbmath.org/?q=an:1071.60003).</cite>