  The connection between the complex Sine and Sinh-Gordon equations on the
complex plane associated with a Weierstrass type system and the possibility of
construction of several classes of multivortex solutions is discussed in
detail. We perform the Painlev\'e test and analyse the possibility of deriving
the B\"acklund transformation from the singularity analysis of the complex
Sine-Gordon equation. We make use of the analysis using the known relations for
the Painlev\'{e} equations to construct explicit formulae in terms of the
Umemura polynomials which are $\tau$-functions for rational solutions of the
third Painlev\'{e} equation. New classes of multivortex solutions of a
Weierstrass system are obtained through the use of this proposed procedure.
  Some physical applications are mentioned in the area of the vortex Higgs
model when the complex Sine-Gordon equation is reduced to coupled Riccati
equations.
