Exact Solutions for a Teleparallel Cosmological Model with Vector Field via Noether Symmetry

We study a cosmological model in the framework of teleparallel gravity, where a vector field [Formula: see text] is non-minimally coupled to the torsion scalar [Formula: see text] in a flat Friedmann-Robertson-Walker (FRW) universe. Using the Noether symmetry approach, we identify specific forms for the coupling function [Formula: see text] and the potential [Formula: see text], with [Formula: see text]. The method allows us to find exact analytical solutions for the scale factor [Formula: see text] and the scalar function [Formula: see text]. General solutions for arbitrary values of [Formula: see text] are derived, but special cases such as [Formula: see text] and [Formula: see text] are studied separately due to their distinct behavior. For [Formula: see text], the model describes a transition from decelerated to accelerated expansion, and depending on the value of the model parameter, a transition to a phantom phase is also possible, which is consistent with phantom dark energy. For the special case [Formula: see text], this model first experiences a deceleration phase and then enters a stable accelerating phase, such that the acceleration parameter [Formula: see text] changes from positive to negative values and the equation of state parameter [Formula: see text] tends to [Formula: see text] at late times, similar to quintessence dark energy. Unlike the case [Formula: see text], in this case, no transition to the phantom phase is observed. Therefore, this model can describe the behavior of the universe during periods of dark energy dominance. In addition, the model is confronted with recent observational data, including SNe Ia, BAO, and CMB through a Bayesian statistical analysis, allowing us to constrain the model parameters. The results indicate that the theoretical predictions are in good agreement with observational data, supporting the viability of the model as a candidate for dark energy.