N = 4 super KdV hierarchy in N = 4 and N = 2 superspaces

We present the results of further analysis of the integrability properties of the N = 4 supersymmetric KdV equation deduced earlier by two of us (F.D. & E.I., Phys. Lett. B 309 (1993) 312) as a hamiltonian flow on N = 4 SU(2) superconformal algebra in the harmonic N = 4 superspace. To make this equation and the relevant hamiltonian structures more tractable, we reformulate it in the ordinary N = 4 and further in N = 2 superspaces. In N = 2 superspace it is represented by a coupled system of evolution equations for a general N = 2 superfield and two chiral and antichiral superfields, and involves two independent real parameters, a and b. We construct a few first bosonic conserved charges in involution, of dimensions from 1 to 6, and show that they exist only for the following choices of the parameters: (i) a = 4, b = 0; (ii) a = −2, b = −6; (iii) a = −2, b = 6. The same values are needed for the relevant evolution equations, including N = 4 KdV itself, to be bi-hamiltonian. We demonstrate that the above three options are related via SU(2) transformations and actually amount to the SU(2) covariant integrability condition found in the harmonic superspace approach. Our results provide a strong evidence that the unique N = 4 SU(2) super KdV hierarchy exists. Upon reduction to N = 2 KdV, the above three possibilities cease to be equivalent. They give rise to the a = 4 and a = −2 N = 2 KdV hierarchies, which thus prove to be different truncations of the single N = 4 SU(2) KdV one.