Using the nonlinear realizations of the Virasoro group we construct the action of the Conformal Quantum Mechanics (CQM) with additional harmonic potential. We show that SL(2, R) invariance group of this action is nontrivially embedded in the reparametrization group of the time which is isomorphic to the centerless Virasoro group. We generalize the consideration to the Ermakov systems and construct the action for the time dependent oscillator. Its symmetry group is also the SL(2, R) ∼ SU (1, 1) group embedded in the Virasoro group in a more complicated way.

Starting from Maxwell-Weyl algebra we found the transformation rules for generalized spacetime coordinates and the differential realization of corresponding generators. By treating local gauge invariance of Maxwell-Weyl group, we presented the Einstein-Cartan-Weyl gravity with the additional terms containing the gauge fields associated with the antisymmetric generators.

Maxwell extension of affine algebra with additional tensorial generators is given. Using the methods of nonlinear realizations, we found the transformation rules for group parameters and corresponding generators. Gauging the Maxwell-affine algebra we presented two possible invariant actions for gravity: one is the first order and the other one is the second order in affine curvature. We noticed that equations of motion for the action, second order in affine curvature, lead to the generalized Bianchi identities on the choice of appropriate coefficients for a particular solution of the constraint equation

In this paper, the Maxwell extension of the special-affine algebra is obtained and corresponding non-linear realization is constructed. We give also the differential realization of the generators of the extended symmetry. Moreover, we present the gauge theory of the Maxwell special-affine algebra and the topological gravity action in four dimensions. As a conclusion, we show that the Bianchi identities can be found by using the solution of the equations of motion.

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