Geodesic
Schlesinger's equations for upper triangular matrices and their solutions
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 64 (2018) no. 1, pp. 86-97

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We consider explicit integral expressions of hypergeometric and hyperelliptic types for solutions of Schlesinger's equations in classes of upper triangular matrices with eigenvalues that produce arithmetic progressions with the same difference. These integral representations extend and generalize earlier known results. 
@article{CMFD_2018_64_1_a5,
     author = {V. P. Lexin},
     title = {Schlesinger's equations for upper triangular matrices and their solutions},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {86--97},
     publisher = {mathdoc},
     volume = {64},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_1_a5/}
}
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V. P. Lexin. Schlesinger's equations for upper triangular matrices and their solutions. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 64 (2018) no. 1, pp. 86-97. http://geodesic.mathdoc.fr/item/CMFD_2018_64_1_a5/