  We construct a one-parameter family of N=3 supersymmetric extensions of the
KdV equation as a Hamiltonian flow on N=3 superconformal algebra and argue that
it is non-integrable for any choice of the parameter. Then we propose a
modified N=3 super KdV equation which possesses the higher order conserved
quantities and so is a candidate for an integrable system. Upon reduction to
N=2, it yields the recently discussed ``would-be integrable'' version of the
N=2 super KdV equation. In the bosonic core it contains a coupled system of the
KdV type equation and a three-component generalization of the mKdV equation. We
give a Hamiltonian formulation of the new N=3 super KdV equation as a flow on
some contraction of the direct sum of two N=3 superconformal algebras.
