The $$\text{m}$$ KdV-type equations related to $$A_5^{(1)}$$ and $$A_5^{(2)}$$ Kac–Moody algebras

We explain the details of the algebraic constructions on nontrivial examples of the mKdV equations related to the $$A_5^{(1)}$$ and $$A_5^{(2)}$$ Kac–Moody algebras. Several types of recursion operators appear naturally in formulating the equations and their Hamiltonian structures. We next introduce the resolvent of the Lax operator and demonstrate that it generates the hierarchy of the Lax representations and also the hierarchy of conservation laws of these equations.