Division algebras and extended N = 2, 4, 8 superKdVs

A no-go result for integrable minimal N = 8 supersymmetric extensions of KdV is found. However, allowing for non-associative realizations of the extended supersymmetries, the first example of an N = 8 supersymmetric KdV equation is explicitly constructed. It involves eight bosonic and eight fermionic fields and corresponds to the unique N = 8 solution based on a generalized Hamiltonian dynamics with (generalized) Poisson brackets given by the non-associative N = 8 superconformal algebra. The complete list of inequivalent classes of parametric-dependent N = 3 and N = 4 superKdVs obtained from the 'non-associative N = 8 SCA' is also furnished. Furthermore, a fundamental domain characterizing the class of inequivalent N = 4 superKdVs based on the 'minimal N = 4 SCA' is given.