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  "body": "\n## Definition\n\nLet $m \\ge 1$ and let $T \\subseteq \\C^m$ be a simply connected domain of points $t = (t_1, \\ldots, t_m)$ with pairwise distinct coordinates.\nFor each $t \\in T$, consider the [[Fuchsian|ZBVH2YE50B]] [[linear system|037ZNY7NAG]]\n$$\\frac{dY}{dz} = \\sum_{i=1}^{m} \\frac{A_i(t)}{z - t_i}\\, Y,$$\nwhere $A_1, \\ldots, A_m \\colon T \\to \\C^{n \\times n}$ are holomorphic.\nFix a base point $z_0$ with $z_0 \\ne t_i$ for all $t \\in T$ and all $i$.\nThe family is an *isomonodromic deformation* if it has a [[fundamental matrix|0H1GN6P9R5]] $Y(z, t)$ with the following property.\n$Y$ is holomorphic in $(z, t)$ for $z$ near $z_0$ and $t \\in T$.\nIts [[monodromy|C8HD62AB6Q]] matrices along loops around the points $t_i$, moved continuously with $t$, do not depend on $t$.\n\n## Remarks\n\nThe family is isomonodromic if the matrices $A_i$ satisfy the Schlesinger equations\n$$\\frac{\\partial A_i}{\\partial t_j} = \\frac{[A_i, A_j]}{t_i - t_j} \\ (j \\ne i), \\qquad \\frac{\\partial A_i}{\\partial t_i} = - \\sum_{j \\ne i} \\frac{[A_i, A_j]}{t_i - t_j},$$\nwhere $[A, B] = A B - B A$.\nFor $2 \\times 2$ systems with singular points $0$, $1$, $t$ and $\\infty$, these equations reduce to the sixth Painlevé equation in the variable $t$.\n\nExample: if all $A_i$ are constant and commute with each other, the Schlesinger equations hold, since all commutators vanish.\nFor $n = 1$, the system is $y' = \\sum_i \\frac{a_i}{z - t_i} y$ with solution $\\prod_i (z - t_i)^{a_i}$, and the monodromy around $t_i$ is $e^{2 \\pi i a_i}$; it does not depend on $t$ when the $a_i$ are constant.\n\nNon-example: for $n = 1$ and $m = 1$, take $a_1(t) = t_1$ on a domain $T \\subseteq \\C$; the monodromy $e^{2 \\pi i t_1}$ around $t_1$ changes with $t_1$, so the family is not isomonodromic.\n",
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