We study a special class of lower trigonometric matrix value boundary problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial hyperbola is given in bilinear form and it shown that only one. Secondly, problem triangular presented transformed into four related problems. Finally, Liouville theorem Painlevé polynomials are used to give solutions.

we consider a mixed boundary value problem for a quarter-plane with a robin condition on one edge. we have developed two procedures, one based on the advanced theory of dual integral equations and the other, in our opinion simpler technique, relying on conformal mapping. both of the procedures are of interest, because the former may be easier to adapt to other boundary value problems.