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Various properties of algebroid solutions of the degenerate third Painlev\'eequation, \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 papercontains connection results for asymptotics as $\tau\to+0$ and as$\tau\to+\infty$ for $a\in\mathbb{C}$. Using these results, the simplestalgebroid 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 basicasymptotic behaviours are visualized.

Algebroid Solutions of the Degenerate Third Painlev\'e Equation for Vanishing Formal Monodromy Parameter