N = 4 Super Kdv Equation

We construct N = 4 supersymmetric KdV equation as a hamiltonian flow on the N = 4 SU(2) super Virasoro algebra. The N = 4 KdV superfield, the hamiltonian and the related Poisson structure are concisely formulated in 1D N = 4 harmonic superspace. The most general hamiltonian is shown to necessarily involve SU(2) breaking parameters which are combined in a traceless rank 2 SU(2) tensor. First nontrivial conserved charges of N = 4 super KdV (of dimensions 2 and 4) are found to exist if and only if the SU(2) breaking tensor is a bilinear of some SU(2) vector with a fixed length proportional to the inverse of the central charge of N = 4 SU(2) algebra. After the reduction to N = 2 this restricted version of N = 4 super KdV goes over to the a = 4 integrable case of N = 2 super KdV and so is expected to be integrable. We show that it is bi-hamiltonian like its N = 2 prototype.