On algebraically integrable differential operators on an elliptic curve (Q431321)

The authors give an exposition of the general theory of algebraically integrable operators (i.e., finite gap), in particular of those on elliptic curves; most of the results here are well known, but for the reader's convenience the authors give an exposition based on differential Galois theory and propose a general conjecture regarding the classification of operators with one pole. They present computational results for third order algebraically integrable operators with one pole and give a conjectural classification of such operators. Finally, algebraically integrable operators of higher order with several poles and with symmetries are studied and a conjecture on the connection with the crystallographic elliptic Calogero-Moser systems is stated which can be seen as a generalization of well-known results of Airault, McKean, and Moser.