readily obtain $\dot{\beta}$ as a function of $P_{\beta}$ and $P_{\phi}$.

What is important is that, after the Legendre transformation, one can get the Lagrangian as follows, $$\begin{aligned}
H=\int_{\Sigma_t} (N{\mathcal H}+N^i{\mathcal H}_i+{\mathcal H}_{matt}+\alpha_NP_N+\alpha_{N^i}P_{N^i}+\alpha_{\chi}P_{\chi})d^3x,\end{aligned}$$ where ${\mathcal H}_{matt}$ is the matter part of the Hamiltonian; $\alpha_N$, $\alpha_{N^i}$ and $\alpha_{\chi}$ are Lagrange multipliers, and the functions ${\mathcal H}$ and ${\mathcal H}_i$ lead to the hamiltonian and diffeomorphism constraints, which come from imposing stability under time evolution of the constraints $P_{N}\approx 0$ and $P_{N^i}\approx0$, i.e., $\dot{P}_N=\{P_N,H\}={\mathcal H}\approx 0$ and $\dot{P}_{N^i}=\{P_{N^i},H\}={\mathcal H}_i\approx 0$.

Now we examine the stability of the constraint $P_{\chi}$. Looking for the only term in the Hamiltonian where $\chi$ appears and for the term where it appears in the action (\[action1\]), i.e. $-N\sqrt{\gamma}(f(\chi)+\beta\chi)$, one has $\dot{P}_{\chi}=\{P_{\chi},H\}=-N\sqrt{\gamma}(f'(\chi)+\beta)$, which leads to the constraint $C_{\chi}\equiv f'(\chi)+\beta \approx 0$. Finally, since $\dot{C}_{\chi}=\{C_{\chi},H\}=N\sqrt{\gamma}f''(\chi)\alpha_{\chi}$, the stability of $C_{\chi}$ is ensured by fixing the Lagrange multiplier $\alpha_{\chi}$ as $\alpha_{\chi}=0$.

Summing up, we have obtained the constraints ${\mathcal H}\cong 0$, ${\mathcal H}_i\cong 0$, $P_{\chi}\cong 0$ and $C_{\chi}\cong 0$, and the canonical pairs $(q_{ij}, P^{ij})$, $(\phi, P_{\phi})$, $(\chi, P_{\chi})$ and $(\beta, P_{\beta})$. Then, from the constraints $P_{\chi}\cong 0$ and $C_{\chi}\cong 0$ one may remove two variables, for example $P_{\chi}\cong 0$ and $\beta\approx 