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We prove that there exists a one-parameter meromorphic solution $u(\tau)$vanishing at $\tau=0$ of the degenerate third Painlev\'e 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)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0,\end{equation*} for formal monodromy parameter $a=\pm i/2$. We studynumber-theoretic properties of the coefficients of the Taylor-series expansionof $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$.These asymptotics are visualized for generic initial data.

One-Parameter Meromorphic Solution of the Degenerate Third Painlev\'{e} Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin