Scalar and vector Keldysh models in the time domain

The exactly solvable Keldysh model of disordered electron system in a random scattering field with extremely long correlation length is converted to the time-dependent model with extremely long relaxation. The dynamical problem is solved for the ensemble of two-level systems (TLS) with fluctuating well depths having the discrete $Z_2$ symmetry. It is shown also that the symmetric TLS with fluctuating barrier transparency may be described in terms of the vector Keldysh model with dime-dependent random planar rotations in $xy$ plane having continuous $SO(2)$ symmetry. Application of this model to description of dynamic fluctuations in quantum dots and optical lattices is discussed.