Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter

Various properties of algebroid solutions of the degenerate third Painlevé equation, \begin{equation*} u^{\prime \prime}(τ) ! = ! \frac{(u^{\prime}(τ))^{2}}{u(τ)} ! - ! \frac{u^{\prime}(τ)}τ ! + ! \frac{1}τ ! \left(-8 \varepsilon (u(τ))^{2} ! + ! 2ab \right) ! + ! \frac{b^{2}}{u(τ)},\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 $\tau \to +\mathrm{\infty }$ for $a\in \mathbb{C}$. Using these results, the simplest algebroid solution with asymptotics $u(\tau )\to c{\tau }^{1\mathrm{/}3}$ as $\tau \to 0$, where $c\in \mathbb{C}\setminus \{0\}$, together with its associated integral ${\int }_0^{\tau }(u(t){)}^{-1}dt$, are considered in detail, and their basic asymptotic behaviours are visualized.