  The dark matter conundrum stands out as one of the central challenges to
understand in both particle physics and cosmology. The ultralight axion-like
particle (UALP) is an appealing dark matter candidate that can be searched for
in neutrino oscillation experiments. In this work, we examined the interaction
among neutrinos and a UALP; such interaction might induce violations of the
Lorentz and $CPT$ symmetries. We assessed the sensitivity to both the isotropic
$CPT-$odd Standard Model Extension (SME) coefficients $(a_{L})^T$ and effective
neutrino-UALP couplings $\tilde{g}$ at the next-to-next generation neutrino
oscillation experiment at the European Spallation Source (ESS).
