New canonical analysis for consistent extension of $\lambda R$ gravity

The canonical analysis of the $\lambda R$ model extended with the term due to Blas, Pujolas, and Sibiryakov $[BPS]$ is performed. The analysis is developed for any value of $\lambda$, but particular attention is paid to the point $\lambda=\frac{1}{3}$ because of the closeness with linearized General Relativity [GR]. Then, we add the higher-order conformal term, the so-called Cotton-square term, to study the constraint structure of what constitutes an example of kinetic-conformal Horava's gravity. At the conformal point, an extra second-class constraint appears; this does not arise at other values of $\lambda$. Then, the Dirac brackets are constructed, and we will observe that the $\lambda R$-Cotton-square model shares the same number of degrees of freedom with linearized $GR$.