The Hamiltonian structure for the supersymmetric $N=2$ Novikov equation is presented. The bosonic sector give us two-component generalization of the cubic peakon equation. The double extended: two-component and two-peakon Novikov equation is defined.
The Bi-Hamiltonian structure for this extended system is constructed.
It is shown that, three different Lax operators in the Dym hierarchy, produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component KdV system. The first equation gives us know
n integrable two-component KdV system while the second reduces to the known symmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations.
It is shown that, two different Lax operators in the Dym hierarchy, produce two generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component KdV system. The first equation gives us new inte
grable two-component KdV system while the second reduces to the known symmetrical two-component KdV equation. For this new two-component coupled KdV system the Lax representation and Hamiltonian structure is defined.
It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densitie
s, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
