The article reproduces the known connection formula for the parameters describing the asymptotic behavior of generic solutions of the SG-PIII equation,
. Similarly to an earlier derivation by Novokshenov, the authors study the direct monodromy problem for a linear ODE associated with SG-PIII. However, they do not apply the conventional WKB analysis of the linear ODE which involves a matching procedure for a Liouville-Green approximation outside a neighborhood of a double turning point and a Weber-Hermit approximation in a vicinity of this double turning point. Instead, the authors use the so-called “uniform asymptotics” method by Clarkson, Bassom, Law and McLeod which extends the Weber-Hermite approximation globally using an appropriate change of variables.