A new integrable sixth-order nonlinear wave equation is discovered by means of the Painlevé analysis, which is equivalent to the Kortewegde Vries equation with a source. A Lax representation and a Bäcklund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.

We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H 0 , H 1 ), where H 0 is a strongly skew-adjoint operator. This is applied to the construction of some new two field integrable systems of PDE by taking the pair (H 0 , H 1 ) in the family of compatible Poisson structures that arose in the study of cohomology of moduli spaces of curves.

We develop a new approach to the Lenard-Magri scheme of integrability of bi-Hamiltonian PDE's, when one of the Poisson structures is a strongly skew-adjoint differential operator.

Non-autonomous Svinolupov -Jordan KdV systems are considered. The integrability criteria of such systems are associated with the existence of recursion operators. A new non-autonomous KdV system is obtained and its recursion operator is given for all N . The examples for N = 2 and N = 3 are studied in detail. Some possible transformations are also discussed which map some systems to autonomous cases.

Abstract. We show that a recently introduced fifth-order bi-Hamiltonian equation with a differentially constrained arbitrary function by A. de Sole, V.G. Kac and M. Wakimoto is not a new one but a higher symmetry of a third-order equation. We give an exhaustive list of cases of the arbitrary function in this equation, in each of which the associated equation is inequivalent to the equations in the remaining cases. The equations in each of the cases are linked to equations known in the literature by invertible transformations. It is shown that the new Hamiltonian operator of order seven, using which the introduced equation is obtained, is trivially related to a known pair of fifth-order and third-order compatible Hamiltonian operators. Using the so-called trivial compositions of lower-order Hamiltonian operators, we give nonlocal generalizations of some higher-order Hamiltonian operators.