What are some known examples of finite-dimensional integrable systems with symplectic Fano phase space?
Here by integrable system we mean a symplectic manifold $(X, \omega)$ of dimension $2n$ with $n$ independent Poisson-commuting functions ('integrals').
The question is whether $X$ admits an almost complex structure $J$ compatible with the symplectic structure (i.e. $\omega(Jv, Ju)=\omega(u, v)$ and $\omega(u, Ju)>0$ for non-zero $u$) such that $\int_{A}c_1(TX)>0$ for homology classes $A \in H_2(X, \mathbb{Z})$ representable by $J$-holomorphic curves (I am quite ignorant of integrable systems with non-Kaehler phase space; that's why I used a symplectic definition without referring to Kaehler structure).
My motivation is largely geometric; namely such an integrable system would endow the phase space with a Lagrangian fibration (Liouville-Arnold).
https://arxiv.org/abs/1605.09736 One example comes from compactified Ruijsenaars-Schneider systems whose phase space is $\mathbb{C}P^n$.
https://arxiv.org/abs/1005.2006 Another example is Gelfand - Zeytlin system whose phase space is flag variety $F_3$.
Some examples of non-K\"ahler symplectic Fano manifolds were constructed by Fine-Panov: https://arxiv.org/abs/0905.3237