  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.) 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 given 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 to be
bi-hamiltonian. We demonstrate that these 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.
