Summary: The Hamiltonian formalism for the Euler equations of an ideal fluid, superconductivity and a barotropic fluid on a \(D\)-dimensional Riemannian manifold is proposed. We show that each of these equations has an infinite series of integrals if \(D\) is even (“generalized entrophies”) and at least one integral if \(D\) is odd (“generalized helicity”). We prove that the magnetic hydrodynamics integral \(\int({\mathbf v}, {\mathbf B})\mu\) is equal to the average linking number of vector fields \(\text{rot }{\mathbf v}\) and \(\mathbf B\) in terms of the ergodic theory. All the invariants considered are Casimir elements (i.e. invariants of coadjoint action) of the corresponding infinite-dimensional Lie algebras.