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In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the super-symmetric Sawada-Kotera (SSK) equation. The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.

A recently proposed three-component Camassa-Holm equation is considered. It is shown that this system is a bi-Hamiltonian system.

We show that the supersymmetric version of two-boson or dispersive water wave system is nothing but one of the systems. We also consider a constrained , which is a gauge equivalence of it.

We construct a new integrable equation which is Miura-related to the cotangent space analog of the Korteweg-de Vries equation. A spectral problem is also given for it.

The constrained Modified KP hierarchy is considered from the viewpoint of modification. It is shown that its second Poisson bracket, which has a rather complicated form, is associated to a vastly simpler bracket via a Miura-type map. Similar results are established for a natural reduction of the MKP.