Bi-Hamiltonian systems from homogeneous operators

Many "famous" integrable systems (KdV, AKNS, dispersive water waves etc.) have a bi-Hamiltonian pair of the following form: $A_1 = P_1 + R_k$ and $A_2 = P_2$, where $P_1$, $P_2$ are homogeneous first-order Hamiltonian operators and $R_k$ is a homogeneous Hamiltonian operator of degree (order) $k$. The Hamiltonian property of $P_1$, $P_2$ and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of $R_k$ was studied in the past (Doyle, Potemin; $k=2,3$) and recently revisited with interesting results.
In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between $P_i$ and $R_k$, $k=2,3$.
See the recent papers arXiv:2602.14739, arXiv:2407.17189, arXiv:2311.13932.
Joint work with P. Lorenzoni and S. Opanasenko.