  Zamolodchikov periodicity is a property of $T$- and $Y$-systems, arising in
the thermodynamic Bethe ansatz. Zamolodchikov integrability was recently
considered as its affine analog in our joint work with P. Pylyavskyy. Here we
prove periodicity and integrability for similar discrete dynamical systems
based on the cube recurrence, also known as the discrete BKP equation. The
periodicity part was conjectured by Henriques in 2007.
