We study the symmetry and integrability of a modified Camassa–Holm equation (MCH), with an arbitrary parameter $k$, of the form $u_t + k(u − u_{xx})^{2}u_x − u_{xxt} + (u^2 − u_x^2)(u_x − u_{xxx} ) = 0.$ The commutator table and adjoint representation of the symmetries are presented to construct one-dimensional optimal system. By using the one-dimensional optimal system, we reduce the order or number of independent variables of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally, we apply the Painlevé test to the resultant nonlinear ordinary differential equation and it is observed that the equation is integrable.

In this paper, we derive the non-isospectral flows of dispersionless modified Kadomtsev–Petviashvili (dmKP) hierarchies by applying quasiclassical limit in the associated Lax equations of the mKP system. Along with the isospectral flows, we investigate the underlying infinite-dimensional Lie algebraic structure of the dmKP system through the construction of implicit flow representations. In addition to this, we also discuss the correspondence between the non-isospectral flows of dKP and dmKP hierarchies by the dispersionless Miura map.