  Various properties of algebroid solutions 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 the monodromy parameter $a=0$ are studied. The paper
contains connection results for asymptotics as $\tau\to+0$ and as
$\tau\to+\infty$ for $a\in\mathbb{C}$. Using these results, the simplest
algebroid solution with asymptotics $u(\tau)\to c\tau^{1/3}$ as $\tau\to0$,
where $c\in\mathbb{C}\setminus\{0\}$, together with its associated integral
$\int_0^\tau {(u(t))^{-1}\,d t}$, are considered in detail, and their basic
asymptotic behaviours are visualized.
