The authors consider an anisotropic Ising model ferromagnet on the honeycomb lattice, having nearest-neighbour pairwise interactions J, theta J>0 along the oblique and horizontal lattice axes, respectively, with theta >0 being an anisotropy parameter. Since the Hamiltonian contains solely z-component (longitudinal) Pauli spin operators, the dynamics in the Ising model evolves only for the non-order (transverse) spin degrees of freedom. An exact solution is calculated for the inelastic neutron scattering function S^xx(q, omega )= Sigma _nA_n(T, theta ) delta ( omega - omega _n( theta )), n=+or-3, +or-2, +or-1, which depends upon the frequency omega , temperature T and anisotropy theta . The discrete energy transfers h(cross) omega _n( theta ) are independent of temperature, vary linearly with theta and possess degeneracies for certain integral values of theta . The scattering amplitudes (intensities/site) A_n(T, theta ) of the Dirac delta -functions exhibit, for fixed theta , such behaviours as rounded maxima, crossing points, and weak energy-type singularities at the critical temperature T_c( theta ). A criticality, the 'spin wave mode' scattering amplitude versus anisotropy displays a shallow asymmetric minimum at theta =1. Compared with the existence of only four inelastic scattering modes in the isotropic ( theta =1) case, the present six modes illustrate the partial lifting of essential degeneracies Some similar results may be inferred for the more general case having two anisotropy parameters and eight modes.