$f(R)$-gravity from Killing Tensors

Understand

We consider $f\left(R\right) $-gravity in a Friedmann-Lema\^itre-Robertson-Walker spacetime with zero spatial curvature.

• We apply the Killing tensors of the minisuperspace in order to specify the functional form of $f\left(R\right) $ and the field equations to be invariant under Lie-B\"acklund transformations which are linear in the momentum (contact symmetries).
• Consequently, the field equations to admit quadratic conservation laws given by Noether's Theorem.
• We find three new integrable $f\left(R\right) $ models, for which with the application of the conservation laws we reduce the field equations to a system of two first-order ordinary differential equations.