Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study

Abstract: A one-parameter family of trans-series asymptotics of solutions to the Degenerate Painlev\'{e} III Equation (DP3E) are parametrised in terms of the monodromy data of an associated two-by-two linear auxiliary problem via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the sigma-forms of the DP3E are also obtained. The actions of Lie-point symmetries for the DP3E are derived.