A (3+1)-dimensional equation admitting a Lax representation

An equation is found that is the second in the hierarchy of higher order equations connected with Dym's equation; equations on the scattering data are introduced; its $(3+1)$-dimensional analogue is considered which admits a Lax representation with the same operator $L$ and an operator $A$ including differentiation with respect to the additional variables; the presence of a countable family of integrable discretizations of the second Korteweg–de Vries equation is demonstrated.