example is the ellipse coming from the holonomy corrected Friedmann equation in standard LQC [@singh06; @svv06; @singh08] (Note that here we are not dealing with the recent model of LQC proposed by Dapor and Liegener in [@dl]) $$\begin{aligned}
\label{flqc}
\rho=\bar g(\bar\chi)=\frac{\rho_c}{2}\left(1\pm\sqrt{1-\frac{4\bar\chi^2}{3\rho_c}}   \right)\Longleftrightarrow H^2=\frac{\rho}{3}\left(1-\frac{\rho}{\rho_c}\right),\end{aligned}$$ where the so-called critical density has the value $\rho_c\sim 0.4 \rho_{pl}=0.4\times 64\pi^2\cong 252$ (see for instance [@singh3]). From this equation one can see that the Hubble parameter vanishes at $\rho=0$ and $\rho=\rho_c$, i.e., the points cutting the axis $H=0$ are $(0,0)$ and $(0,\rho_c)$, as one can see from Figure $1$.

![Ellipse depicted by the Friedmann equation in standard LQC, and its dynamics for a non-phantom fluid with linear Equation of State (EoS) $P=w\rho$.](HrhoLQC.png)

For this background, using equation (\[formula\]), the corresponding $f$-theory is given by $$\begin{aligned}
\label{fLQC}
\bar f(\bar \chi)=\frac{1}{3}\bar\chi^2+\frac{\rho_c}{2}\left(1-\sqrt{1-s^2}-s\arcsin s  \right),\end{aligned}$$ where $s\equiv \frac{2}{\sqrt{3\rho_c}}\bar\chi$ and the functions $\sqrt{1-s^2}$ and $\arcsin s$ are bi-valued [@hp18]. For the theory to be well defined, since the ellipse has two branches -the upper part corresponding to $\rho=\frac{\rho_c}{2}\left(1+\sqrt{1-\frac{4\bar\chi^2}{3\rho_c}} \right)$ and the lower one $\rho=\frac{\rho_c}{2}\left(1-\sqrt{1-\frac{4\bar\chi^2}{3\rho_c}} \right)$-, we have to choose a convenient prescription. For example, we can choose the sign of the square root as positive (respectively negative) in the lower (respectively 