 Square root of a CIR diffusion

For $d{C}_t=a(b-C_t)dt+\sigma \sqrt{C_t}d{W}_t$, the Itô formula applied to $Z=\sqrt{C}$ gives the displayed dynamics before hitting zero. In particular $2ab={\sigma }^2$ does not eliminate the reciprocal drift. The cancellation condition is $4ab={\sigma }^2$. Under that condition $C$ can be represented as the square of a signed Ornstein-Uhlenbeck process with drift $-aY/2$ and volatility $\sigma /2$; its principal square root is the reflected process $|Y|$, not an unrestricted signed OU process. The mean of $C$ is $b+(C_0-b)e^{-at}$ for all admissible positive parameters.