# §32.4 Isomonodromy Problems

## §32.4(i) Definition

can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose

is a linear system in which and are matrices and is independent of . Then the equation

32.4.2

is satisfied provided that

32.4.3

(32.4.3) is the compatibility condition of (32.4.1). Isomonodromy problems for Painlevé equations are not unique.

## §32.4(ii) First Painlevé Equation

is the compatibility condition of (32.4.1) with

32.4.4
32.4.5

## §32.4(iii) Second Painlevé Equation

is the compatibility condition of (32.4.1) with

32.4.6
32.4.7

See Flaschka and Newell (1980).

## §32.4(iv) Third Painlevé Equation

The compatibility condition of (32.4.1) with

32.4.8
32.4.9

where is an arbitrary constant, is

32.4.10
32.4.11
32.4.12
32.4.13

If , then

32.4.14

and satisfies  with

where

32.4.16

Note that the right-hand side of the last equation is a first integral of the system (32.4.10)–(32.4.13).

## §32.4(v) Other Painlevé Equations

For isomonodromy problems for , , and  see Jimbo and Miwa (1981).