The odd integrable system

In a volume dedicated to Yuri Manin on his 50th birthday the speaker introduced an integrable system defined on the cotangent bundle of the moduli space of stable $G$-bundles

over an algebraic curve. The talk concerns an odd version of this: commuting polyvector fields on these moduli spaces defined by bi-invariant alternating forms on the Lie algebra of $G$. We shall focus on concrete examples and raise questions about higher cohomology groups.