One-parameter meromorphic solution of the degenerate third Painlevé equation with formal monodromy parameter $a=\pm\mathrm{i}/2$ vanishing at the origin

We prove that there exists a one-parameter family of meromorphic solutions $u(\tau)$ vanishing at $\tau=0$ of the degenerate third Painlevé equation,
\begin{equation*} u^{\prime \prime}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau} \left(-8 \varepsilon (u(\tau))^{2} + 2ab \right) + \frac{b^{2}}{u(\tau)},\ \varepsilon=\pm1,\ \varepsilon b>0, \end{equation*}
for formal monodromy parameter $a=\pm\mathrm{i}/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$. These asymptotics are visualized for generic initial data.