In this case, the conservation equation has the following analytic solution $$\label{solution}
\bar{\phi}=\frac{2}{\sqrt{3}}\ln\left( \sqrt{\frac{3}{4}\rho_c}t+\sqrt{\frac{3}{4}\rho_ct^2+1}  \right),$$ leading to the following background $$\begin{aligned}
\label{background}
a(t)=\left(\frac{3}{4}\rho_ct^2+1  \right)^{1/3}, \quad H=\frac{\frac{1}{2}\rho_ct}{\frac{3}{4}\rho_ct^2+1 }, \quad \rho=\frac{\rho_c}{\frac{3}{4}\rho_ct^2+1 },
$$ which is the same nonsigular bouncing background obtained solving the equation (\[LQCstandard\]) when one considers a pressureless fluid and whose dynamics is depicted in Figure $1$. All the other solutions, which lead to different backgrounds, are obtained choosing different initial conditions (see Figure $3$ of [@ha17], where it is showed that nearly all solutions lead to nonsingular bouncing backgrounds).

Moreover, since we are dealing with theories beyond GR, in order to have a bouncing background it is not needed to violate the [*null energy condition*]{} $\rho+P\geq 0$ near the bounce using quintom or Lee-Wick matter (see [@easson] and references therein) because the bounce occurs when the value of the energy density is strictly positive. For example, looking at (\[background\]) one can see that at the bouncing time $t=0$ the energy density is given by $\rho_c$ and the pressure, which could be obtained from the Raychaudhuri equation (\[LQCstandard\]), is zero. So, the null energy condition is fulfilled at the bounce.

Hamiltonian analysis
====================

In this section we perform a Hamiltonian 