  New exact completely closed homogeneous Generalized Master Equations (GMEs),
governing the evolution in time of equilibrium two-time correlation functions
for dynamic variables of a subsystem of s particles (s<N) selected from N>>1
particles of a classical many-body system, are obtained These time-convolution
and time-convolutionless GMEs differ from the known GMEs (e.g. Nakajima-Zwanzig
GME) by absence of inhomogeneous terms containing correlations between all N
particles at the initial moment of time and preventing the closed description
of s-particles subsystem evolution. Closed homogeneous GMEs describing the
subdynamics of fluctuations are obtained by applying a special projection
operator to the Liouville equation governing the dynamics of N-particle system.
In the linear approximation in the particles' density, the linear Generalized
Boltzmann equation accounting for initial correlations and valid at all
timescales is obtained This equation for a weak inter-particle interaction
converts into the generalized linear Landau equation in which the initial
correlations are also accounted for. Connection of these equations to the
nonlinear Boltzmann and Landau equations are discussed.
