The modified gravity models with higher derivatives with respect to scalar curvature can be transformed to GR with a few scalar fields using Lagrange multipliers and a conformal transformation from Jordan to Einstein frame. Such resulting models can be presented as Chiral Self-Gravitating Models with fixed functional dependence for a chiral (target) space and the potential energy.
In the present contribution, we study Killing symmetries for the chiral spaces corresponding to f(R, (\nabla R)^2), f(R, \Box R) and few versions of f(R, (\nabla R)^2, \Box R) gravity. Special investigation is devoted to the modified f(R) gravity with a kinetic scalar curvature of the form: f(R, (\nabla R)^2, \Box R)=f_1(R) +X(R) \nabla_\mu R \nabla^\mu R. We investigate connection of obtained Killing vectors of target space with Killing symmetry of Friedmann-Robertson-Walker and spherically symmetric spacetimes with the aim to find exact solutions of the models under consideration.
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