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  "body": "\n## Summary\n\nAn isomonodromic deformation is a family of linear differential equations whose [[monodromy|C8HD62AB6Q]] stays the same.\nThe family depends on parameters, such as the positions of the singular points.\nKeeping the monodromy fixed forces the coefficients to satisfy nonlinear differential equations in the parameters.\nFor Fuchsian systems these are the Schlesinger equations, and in the simplest nontrivial case they reduce to the sixth Painlevé equation.\nThis gives one explanation of why the Painlevé equations have the [[Painlevé property|8EV1G2Y6CC]], and it links them to random matrices and integrable systems.\nSee the [[definition|SVDH9C5MCK]].\n\n## Explanations\n\n### L1 human\n\nTake a linear equation with a few singular points and move the singular points a little.\nIn general the monodromy, the way solutions get mixed around the points, changes.\nOne can ask to move the points and adjust the coefficients at the same time so that the monodromy does not change.\nThe coefficients then follow a nonlinear equation in the positions of the points.\nMany famous nonlinear equations arise this way.\n",
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