The method of isomonodromic deformations for the “degenerate” third Painleve equation

In order to investigate solutions of the equation $(\tau u_\tau)_\tau=e^u-e^{-2u}$, which is a variant of the “degenerate” third Painleve equation, some linear differential equation in $3\times3$ matrices is considered. We parametrize asymptotics of solutions of the nonlinear Painleve equation at $\tau\to0$ as well as asymptotics of the regular solutions at $\tau\to\pm\infty$ in terms of the monodromy data of the linear equation.