Isomonodromic deformation

Remarks

The family is isomonodromic if the matrices $A_i$ satisfy the Schlesinger equations $$\frac{\partial A_i}{\partial t_j} = \frac{[A_i, A_j]}{t_i - t_j} \ (j \ne i), \qquad \frac{\partial A_i}{\partial t_i} = - \sum_{j \ne i} \frac{[A_i, A_j]}{t_i - t_j},$$ where $[A, B] = A B - B A$. For $2 \times 2$ systems with singular points $0$, $1$, $t$ and $\infty$, these equations reduce to the sixth Painlevé equation in the variable $t$.

Example: if all $A_i$ are constant and commute with each other, the Schlesinger equations hold, since all commutators vanish. For $n = 1$, 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.

Non-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.