We study the Camassa-Holm (CH) equation and recently introduced

The modern theory of integrable nonlinear partial differential equations arose as a result of the inverse scattering method (ISM) discovered by Gardner et al. [

Sasaki [

Let us recall some facts about the equations we study in this paper.

The Camassa-Holm equation (CH)

The bi-Hamiltonian form of (

There exists an infinite sequence of conservation laws (multi-Hamiltonian structure)

The

The bi-Hamiltonian form of (

Recently in [

Kupershmidt [

In fact, Kersten et al. [

The aim of this paper is to show that the CH and the

As a matter of fact, Yao and Zeng [

Kundu et al. [

Guha [

The paper is organized as follows. In Section

In this section, we recall some definitions and facts. One can consult, for example, [

A scalar differential equation

Equations (

An equation of pseudospherical type is the integrability condition for a

An equation

Hence, if

Another important property of equations of pseudospherical type is that they admit quadratic pseudopotentials. Pseudopotentials are a generalization of conservation laws.

Let

Geometrically, Pfaffian systems (

In this section, we consider the nonholonomic deformation of CH equation and

Recall from Introduction the CH equation (

Following Kupershmidt's construction, we introduce the nonholonomic deformation of the CH equation

The system (

Let us give the corresponding 1-forms

For the proof of Proposition

For the matrices

In order to apply Proposition

The nonholonomic deformation of the CH equation (

Conservation densities can be obtained by expanding (

Consider now the

The nonholonomic deformation of the

For validation of Proposition

For the matrices

In order to find pseudopotentials for the nonholonomic deformation of the

The nonholonomic deformation of the

As the conserved densities for the nonholonomic deformation are the same as for the original bi-Hamiltonian system, we make use of the pseudopotentials to obtain them for the

We finish this section with the geometric integrability of one of the most popular two-component generalization of CH equation and of KdV6 equation.

Another generalization of the Camassa-Holm equation is the following integrable two-component CH system [

Finally, we note that nonholonomic perturbation of KdV equation, known as KdV6 equation, is also of pseudospherical type, that is, KdV6 equation is geometrically integrable. We just give the corresponding 1-forms

In this paper, we study the CH equation and some of its generalizations from the geometric point of view. We show that Kupershmidt deformations for CH and

Having at hand these examples of geometrically integrable Kupershmidt deformations, it is natural to think that maybe there exists a general link in this sense: a Kupershmidt deformation of geometrically integrable system is again geometrically integrable. We have not succeeded in establishing such a link up to now, but we believe that this is true at least for the systems with local Hamiltonian pair of operators as in the above examples.

Let us return, however, to the

Denote by

We have

This work is partially supported by Grant no. 169/2010 of Sofia University and by Grant no. DD VU 02/90 with NSF of Bulgaria.