  The magnetic structures and spin dynamics of Na$_{2}$Co$_{2}$TeO$_{6}$ and
Na$_{2}$Ni$_{2}$TeO$_{6}$ are investigated by means of elastic and inelastic
neutron scattering measurements and the results are discussed in the context of
a generalized Kitaev-Heisenberg model on honeycomb lattice with strong
spin-orbit coupling. The large number of parameters involved in the Hamiltonian
model are evaluated by using an iterative optimization algorithm capable of
extracting model solutions and simultaneously estimating their uncertainty. The
analyses establish that both Co$^{2+}$ ($d^7$) and Ni$^{2+}$ ($d^8$)
antiferromagnets realize bond-dependent anisotropic nearest-neighbor
interactions, and support the theoretical predictions for the realization of
Kitaev physics in 3$d$ electron systems with effective spins $S$=1/2 and $S$=1.
Furthermore, by studying the Na-doped system Na$_{2.4}$Ni$_{2}$TeO$_{6}$, we
show that the control of Na content can provide an effective route for fine
tuning the magnetic lattice dimensionality, as well as to controlling the
bond-dependent anisotropic interactions.
