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.
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