
## 1. Painlevé Property

1. #### Problem 1.1.

Given an equation with a Painlevé property, how does one identify which linear system it is?
• #### Problem 1.2.

Can one solve the above for discrete Painlevé? (If one has a nonlinear difference equation that one suspects may be Painlevé, can one work out which it is?)
• #### Problem 1.3.

Can one do 1.1 for discrete Painlevé?
• #### Problem 1.4.

How does one check the Painlevé property?
1. Remark. [Peter Clarkson] Use the Ablowitz-Ramani-Segur algorithm [M.J. Ablowitz, A. Ramani & H. Segur, "A connection between nonlinear evolution equations and ordinary differential equations of P-type. I", J. Math. Phys., 21 (1980), 1006-1015]
• #### Problem 1.5.

In general, is there a classification of determinants that matches that for Painlevé?
• #### Problem 1.6.

Are the spacing distributions of the exceptional groups related to Painlevé?
Answer in Chapter 8 of P. Forrester
• #### Problem 1.7.

Is there an integrable structure related to general $\beta$-ensembles?

Cite this as: AimPL: Painleve equations and their applications, available at http://aimpl.org/painleveapp.