Hitchin and Beauville-Mukai System: Classical and Quantum Correspondence

This is a brief review of my recent works with B. Enriquez on the relations between Hitchin and Beauville-Mukai algebraically completely integrable systems (ACIS) with emphasis on the quantization of their birational correspondence. The second part of the contribution gives an example of the construction specifying by its relation with Sklyanin-Feigin-Odesskii elliptic algebras and based on my works with A. Odesskii. This contribution is a content of two my talks presented at the Twente Conferences on Lie Groups in December 2004 and December 2005.