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.

It seems natural to view the Itô process on a different time scale where it becomes a Itô diffusion, and in turn, invoke existence/uniqueness theory for Itô diffusion.

In particular, under the (deterministic) time change given by $\tau(t) = (t-2 \alpha t )^{\frac{1}{1-2 \alpha}}$ (so that   $\tau'(t) = \tau(t)^{2 \alpha}$ and $\tau(0)=0$), the time-changed process is (in law) an Itô diffusion that satisfies $$ d \tilde{X}_t = b(\tilde{X}_t) d B_t $$ where $\tilde{X}_t = X_{\tau(t)}$ and $B_t=\int_0^{\tau(t)} s^{-\alpha} dW_s$ is again a standard Brownian motion. Note that this time change only makes sense for $\alpha \in [0,1/2)$. 

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.

For more about the correspondence between an Itô diffusion and Itô process via time change, see

Øksendal, Bernt, When is a stochastic integral a time change of a diffusion?, J. Theor. Probab. 3, No. 2, 207-226 (1990). ZBL0698.60046.