  We introduce and study the dynamics of Chebyshev polynomials on $d>2$ real
intervals. We define isoharmonic deformations as a natural generalization of
the Chebyshev dynamics. This dynamics is associated with a novel class of
constrained isomonodromic deformations for which we derive the constrained
Schlesinger equations. We provide explicit solutions to these equations in
terms of differentials on an appropriate family of hyperelliptic curves of any
genus $g=d-1\ge 2$. The verification of the obtained solutions relies on the
combinatorial properties of the Bell polynomials and on the analysis on the
Hurwitz spaces. From the point of view of the classical algebraic geometry we
formulate and solve the problem of constrained Jacobi inversion for
hyperelliptic curves. We discuss applications of the obtained results in
integrable systems, e.g. billiards within ellipsoids in $\mathbb R^d$.
