Henrik Aratyn

The paper discusses P _III–V equation for special values of its parameters for which this equation reduces to P _III , I₁₂, as well as, to some special cases of I₃₈ and I₄₉ equations from the Ince's list of 50 second order differential equations possessing Painlevé property. These reductions also yield symmetries governing the reduced models obtained from the P _III–V equation. We point out that the solvable equations on Ince's list emerge in this reduction scheme when the underlying reflections of the Weyl symmetry group no longer include an affine reflection through the hyperplane orthogonal to the highest root and therefore do not give rise to an affine Weyl group. We hypothesize that on the level of the underlying algebra and geometry this might be a fundamental feature that distinguishes the six Painleve equations from the remaining 44 solvable equations on the Ince's list.

We discuss symmetries of Hamiltonians of I₃₈ and I₄₉ equations that appear on Ince's list of fifty second-order differential equations with Painlevé property. This study is informed by structure of Weyl symmetries of Painlevé P _III and mixed Painlevé P _III–V equations and provides insights into differences between the symmetries of Painlevé equations and symmetries of solvable equations on Ince's list.

We review the construction of the mixed Painlevé P system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of a hybrid differential equation that for special limits of its parameters reduces to either Painlevé P _III or P _V . The aim of this paper is to describe solutions of the P model. In particular, we determine and classify rational, power series and transcendental solutions of the P model. A class of power series solutions is shown to be convergent in accordance with the Briot–Bouquet theorem. Moreover, the P equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of a P equation is quantized by integer .

A mixed symmetric Painlevé III–V model which describes a hybrid of both equations is defined and obtained by successive self-similarity and Dirac Lagrange multiplier reductions from an integrable 4-boson hierarchy.