Isoperiodic deformations of Toda curves and chains, the difference Korteweg - de Vries equation, and $SU(N)$ Seiberg-Witten theories

We introduce the dynamics of Toda curves of order $N$ and derive differential equations governing this dynamics. We apply the obtained results to describe isoperiodic deformations of $N$-periodic Toda chains and periodic difference Korteweg-de Vries equation. We describe deformations of the essential spectra of $N$-periodic two-sided Jacobi matrices. We also study singular regimes of $SU(N)$ Seiberg-Witten theory and describe their deformations preserving the number of singularities where new massless particles may occur. We introduce and describe isoequilibrium deformations of arbitrary collections of $d$ real disjoint closed intervals. We conclude by providing explicit triangular solutions to constrained Schlesinger systems.