Abstract
We set out a method for giving explicit algebraic coordinates on varieties of elliptic solitons, which consists of finding the spectral curve by elimination and solving the Jacobi inversion problem by the use of Kleinian functions and their identities. As an example we solve a five-particle elliptic Calogero-Moser system, whose spectral curve turns out to be completely reducible over three equianharmonic elliptic curves.