Non-Universal Flipped Trinification Models with Arbitrary $\beta$

We explore the recently proposed gauge symmetry $SU(3)_C \otimes SU(3)_L \otimes SU(3)_R \otimes U(1)_X$, which naturally embeds both the Left-Right symmetric model and the 3-3-1 model as subgroups. Within this unified framework, we propose four families of leptons and quarks. A detailed analysis of their contributions to gauge anomaly cancellation is carried out for a general value of the parameter $\beta$. From this analysis, eight non-universal anomaly-free three-family models and four non-universal two-family anomaly free sets were identified. The three-family models offer realistic extensions of the Standard Model, retaining several appealing features of the 3-3-1 models, while the two-family sets provide flexibility for constructing models with even numbers of families. We also report LHC bounds on the $Z'$ mass for the particular case $\beta = -1/\sqrt{3}$, considering all possible combinations of lepton and quark families. These limits exhibit a strong dependence on the mixing parameter $\theta$, which enters the couplings of Standard Model fermions to the $Z'$ boson.