  In this paper we classify symplectic leaves of the regular part of the
projectivization of the space of meromorphic endomorphisms of a stable vector
bundle on an elliptic curve, using the study of shifted Poisson structures on
the moduli of complexes from our previous work \cite{HP17}. This Poisson
ind-scheme is closely related to the ind Poisson-Lie group associated to
Belavin's elliptic $r$-matrix, studied by Sklyanin, Cherednik and Reyman and
Semenov-Tian-Shansky. Our result leads to a classification of symplectic leaves
on the regular part of meromorphic matrix algebras over an elliptic curve,
which can be viewed as the Lie algebra of the above-mentioned ind Poisson-Lie
group. We also describe the decomposition of the product of leaves under the
multiplication morphism and show the invariance of Poisson structures under
autoequivalences of the derived category of coherent sheaves on an elliptic
curve.
