KdV6: An Integrable System

Understand

$K^2 S^2 T [5]$ recently derived a new 6$^{th}$-order wave equation $KdV6$: $(\partial^2_x + 8u_x \partial_x + 4u_{xx})(u_t + u_{xxx} + 6u_x^2) = 0$, found a linear problem and an auto-B${\ddot{\rm{a}}}$ckclund transformation for it, and conjectured its integrability in the usual sense.

• We prove this conjecture by constructing an infinite commuting hierarchy $KdV_n6$ with a common infinite set of conserved densities.
• A general construction is presented applicable to any bi-Hamiltonian system (such as all standard Lax equations, continuous and discrete) providing a nonholonomic perturbation of it.
• This perturbation is conjectured to preserve integrability.