We extend Painlevé IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlevé IV and II equations for special limits of the underlying parameters. We derive the underlying Bäcklund transformations, symmetry structure and requirements to satisfy Painlevé property.

The paper discusses PIII−V equation for special values of its parameters for which this equation reduces to PIII , 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 PIII−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 Painlevé equations from the remaining 44 solvable equations on the Ince's list.

We identify the self-similarity limit of the second flow of sl(N) mKdV hierarchy with the periodic dressing chain thus establishing a connection to A N − 1 ( 1 ) invariant Painlevé equations. The A N − 1 ( 1 ) Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra s l ̂ ( N ) endowed with a principal gradation.

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 PIII and mixed Painlevé PIII−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 III−V system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an 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 P III−V model. In particular, we determine and classify rational, power series and transcendental solutions of P III−V . A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P III−V 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 P III−V equation is quantized by integer n ∈ Z.Painlevé P II equation. Painlevé P V equations appeared in the context of impenetrable Bose gas model [13]. More recently the scattering on two Aharonov-Bohm vortices was exactly solved in terms of solutions of the Painlevé III equation [7]. Ablowitz Ramani and Segur (ARS) showed a connection between the mKdV integrable model and the Painlevé P II equation when self-similarity limit was implemented. They suggested that such connection could be generalized to other integrable models [1] meaning that the self-similarity limit would lead for integrable models to equations with the Painlevè property. More recently, it was found [3] that the 2n boson integrable model obtained as particular reductions (Drinfeld-Sokolov) of KP integrable models connected to Toda lattice hierarchy gives rise to higher Painlevé equations invariant under extended affine Weyl groups. In particular, investigation of various Dirac reduction schemes applied to the 4-boson (n = 2) integrable model with Weyl symmetry structure A (1) 4 led to emergence of a new mixed P III−V model [4]. The second order P III−V equation for a canonical variable q is given by: