  We study the homogeneous turbulence in the presence of a constant average
velocity gradient in an infinite fluid domain, with a novel finite-scale
Lyapunov analysis, presented in a previous work dealing with the homogeneous
isotropic turbulence.
  Here, the energy spectrum is studied introducing the spherical averaged pair
correlation function, whereas the anisotropy caused by the velocity gradient is
analyzed using the equation of the two points velocity distribution function
which is determined through the Liouville theorem. As a result, we obtain the
evolution equation of this velocity correlation function which is shown to be
valid also when the fluid motion is referred with respect to a rotating
reference frame. This equation tends to the classical von K\'arm\'an-Howarth
equation when the average velocity gradient vanishes.
  We show that, the steady energy spectrum, instead of following the Kolmogorov
law $\kappa^{-5/3}$, varies as $\kappa^{-2}$. Accordingly, the structure
function of the longitudinal velocity difference $<\Delta u_r^n> \approx
r^{\zeta_n}$ exhibits the anomalous scaling $\zeta_n \approx n/2$, and the
integral scales of the correlation function are much smaller than those of the
isotropic turbulence.
