Complex Sine-Gordon-2: A New Algorithm for Multivortex Solutions on the Plane

We present a new vorticity-raising transformation for the second integrable complexification of the sine-Gordon equation on the plane. The new transformation is a product of four Schlesinger maps of the Painleve-V equation to itself and allows constructing the $n$-vortex solution more efficiently than the previously reported transformation comprising a product of $2n$ maps.