Two-parameter families of MPO integrals of motion in Heisenberg spin chains

Abstract: Recently, Fendley et al. (2025) [arXiv:2511.04674] revealed a new way to demonstrate the integrability of XYZ Heisenberg model by constructing a one-parameter family of integrals of motion in the matrix product operator (MPO) form. In this short note, I report on the discovery of two-parameter families of MPOs that commute with with the Heisenberg spin chain Hamiltonian in the XXX, XXZ, and XYZ cases. I describe a symbolic algebra approach for finding such integrals of motion and speculate about possible applications.