Inviscid, incompressible hydrodynamics and incompressible ideal magnetohydrodynamics MHD share many properties such as time-reversal invariance of equations, conservation laws, and certain topological features. In three dimensions, these systems may lead to singular solutions involving vortex and current sheets. While dissipative viscoresistive effects can regularize the equations leading to bounded solutions to the initial-boundary value Cauchy problem which presumably exist uniquely, the time-reversal symmetry and associated conservation properties are certainly destroyed. The present work is analogous to and suggested by the Korteweg–de Vries regularization of the one-dimensional, nonlinear kinematic wave equation. Thus the regularizations applied to the original equations of hydrodynamics and ideal MHD retain conservation properties and the symmetries of the original equations. Integral invariants which generalize those known for the original systems are shown to imply bounded enstrophy. The regularization developed can also be applied to the corresponding dissipative models such as the Navier–Stokes equations and the viscoresistive MHD equations and may imply interesting regularity properties for the solutions of the latter as well. The models developed thus have intrinsic mathematical interest as well as possible applications to large-scale numerical simulations in systems where dissipative effects are extremely small or even absent.