On a Generalized Camassa–Holm Equation Describing Pseudospherical Surfaces

In this paper, a new integrable Camass–Holm type equation with cubic nonlinearity is studied. The Lie symmetries and group-invariant solutions of the equation are presented. Geometrically, this equation describes a non-trivial one-parameter family of pseudo-spherical surfaces and is the integrability condition of an $$\mathfrak {sl}(2, \mathbb {R})$$-valued linear problem. Moreover, its quadratic pseudo-potentials and infinite number of conservation laws are directly constructed. In addition, the equation is proved to possesses the property of admitting local isometric immersions in $$\mathbb {E}^3$$ with the “universal" coefficients of the second fundamental form, which depend only on x and t.