It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:
Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.
If $d\neq 0$, the system is not Fuchsian: singularity at $\infty$ is irregular. If $k=2, d\neq 0$ your system already contains the "prolate/oblate spheroid equations", for which were studied much and no reasonable explicit formula for the monodromy is known.