An extension of the modified Sawada–Kotera equation and conservation laws

Based on the modified Sawada--Kotera equation, we introduce a 3?3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawada--Kotera equation, by which a Lax pair of the modified Sawada--Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawada--Kotera equation, the Kaup--Kupershmidt equation, and the modified Sawada--Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada--Kotera equation are constructed with the aid of their Lax pairs.