  For multipartite states we consider a notion of D-symmetry. For a system of
$N$ qubits it concides with usual permutational symmetry. In case of $N$ qudits
($d\geq 3$) the D-symmetry is stronger than the permutational one. For the
space of all D-symmetric vectors in $(\mathbb{C}^d)^{\otimes N}$ we define a
basis composed of vectors $\{|R_{N,d;k}\rangle: \,0\leq k\leq N(d-1)\}$ which
are analog for Dicke states. The aim of this paper is to discuss the problem of
separability of D-symmetric states which are diagonal in the basis
$\{|R_{N,d;k}\rangle\}$. We show that if $N$ is even and $d\geq 2$ is arbitrary
then a PPT property is necessary and sufficient condition of separability for
D-invariant diagonal states. In this way we generalize results obtained by Yu
for qubits. Our strategy is to use some classical mathematical results on a
moment problem.
