  We study the Schlesinger system of partial differential equations in the case
when the unknown matrices of arbitrary size $(p\times p)$ are triangular and
the eigenvalues of each matrix form an arithmetic progression with a rational
difference $q$, the same for all matrices. We show that such a system possesses
a family of solutions expressed via periods of meromorphic differentials on the
Riemann surfaces of superelliptic curves. We determine the values of the
difference $q$, for which our solutions lead to explicit polynomial or rational
solutions of the Schlesinger system. As an application of the
$(2\times2)$-case, we obtain explicit sequences of rational solutions and
one-parameter families of rational solutions of Painlev\'e VI equations. Using
similar methods, we provide algebraic solutions of particular Garnier systems.
