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We study differential operators on an elliptic curve of order higher than 2which are algebraically integrable (i.e., finite gap). We discussclassification of such operators of order 3 with one pole, discovering exoticoperators on special elliptic curves defined over ${\mathbb Q}$ which do notdeform to generic elliptic curves. We also study algebraically integrableoperators 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).

On Algebraically Integrable Differential Operators on an Elliptic Curve