Three-dimensional hydrodynamic turbulence is investigated under the assumptions of homogeneity and weak axisymmetry. Following the kinematics developed by E. Lindborg [J. Fluid Mech. 302, 179 (1995)] we rewrite the von Kármán-Howarth equation in terms of measurable correlations and derive the exact relation associated with the flux conservation. This relation is then analyzed in the particular case of turbulence subject to solid-body rotation. We make the ansatz that the development of anisotropy implies an algebraic relation between the axial and the radial components of the separation vector r and we derive an exact vectorial law which is parametrized by the intensity of anisotropy. A simple dimensional analysis allows us to fix this parameter and find a unique expression.