§32.4 Isomonodromy Problems

${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ 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 $𝐀$ and $𝐁$ are matrices and $\lambda$ is independent of $z$. Then the equation

is satisfied provided that

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

${\text{P}}_{\text{I}}$ is the compatibility condition of (32.4.1) with

${\text{P}}_{\text{II}}$ is the compatibility condition of (32.4.1) with

See Flaschka and Newell (1980).

The compatibility condition of (32.4.1) with

where ${\theta }_{\mathrm{\infty }}$ is an arbitrary constant, is

If $w=-u_0/(v_0{v}_1)$, then

and $w$ satisfies ${\text{P}}_{\text{III}}$ with

where

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

For isomonodromy problems for ${\text{P}}_{\text{IV}}$, ${\text{P}}_{\text{V}}$, and ${\text{P}}_{\text{VI}}$ see Jimbo and Miwa (1981).