  We study differential operators on an elliptic curve of order higher than 2
which are algebraically integrable (i.e., finite gap). We discuss
classification of such operators of order 3 with one pole, discovering exotic
operators on special elliptic curves defined over ${\mathbb Q}$ which do not
deform to generic elliptic curves. We also study algebraically integrable
operators of higher order with several poles and with symmetries, and
(conjecturally) relate them to crystallographic elliptic Calogero-Moser systems
(which is a generalization of the results of Airault, McKean, and Moser).
