On the properties of solutions of a system of two nonlinear differential equations associated with the Josephson model

We investigate the analytic properties of solutions of a system of two first-order nonlinear differential equations with an arbitrary parameter $l$ associated with an overdamped Josephson model. We reduce this system to a system of differential equations that is equivalent to the fifth Painlevé equation with the sets of parameters
$$\biggl(\frac{(1-l)^2}{8}, -\frac{(1-l)^2}{8},0,-2\biggr), \; \biggl(\frac{l^2}{8}, -\frac{l^2}{8},0,-2\biggr).$$
We show that the solution of the third Painlevé equation with the parameters $(-2l, 2l-2,1,-1)$ can be represented as the ratio of two linear fractional transformations of the solutions of the fifth Painlevé equation (with the parameters in the above sequence) connected by a Bäcklund transformation.