  We investigate the correlators of three single-trace operators in
Aharony-Bergman-Jafferis-Maldacena (ABJM) theory from the perspective of
integrable boundary states. Specifically, we focus on scenarios where two
operators being $1/3$-BPS and the entire correlation function is considered
within the twisted-translated frame. The correlator can be expressed as the
overlap between a boundary state and a Bethe state. It is found that the
boundary state formed by the two $1/3$-BPS operators is integrable only when
the number of Wick contractions between the non-BPS operator and one of the
$1/3$-BPS operators is 0 or 1. We compute the overlaps for the integrable cases
utilizing the symmetries preserved by the correlators.
