On integrability of a new dynamical system associated with the BBM‐type hydrodynamic flow

This article explores the exceptional integrability property of a family of higher‐order Benjamin –Bona–Mahony (BBM)‐type nonlinear dispersive equations. Here, we highlight its deep relationship with a generalized infinite hierarchy of the integrable R iemann ‐type hydrodynamic equations. A previous L ie symmetry analysis revealed a particular case which was conjectured to be integrable. Namely, a Lie –Bäcklund symmetry exists, thus highlighting another associated dynamical system. Here, we investigate these two equations using the gradient‐holonomic integrability scheme. Moreover, we construct their infinite hierarchy of conservation laws analytically, using three compatible P oisson structures to prove the complete integrability of both dynamical systems. We investigate these two equations using the so‐called gradient‐holonomic integrability scheme. Based on this scheme, applied to the equation under consideration, we have analytically constructed its infinite hierarchy of conservation laws, derived two compatible P oisson structures, and proved its complete integrability.