  We consider a $10$-dimensional family of Lehn-Lehn-Sorger-van Straten
hyperk\"ahler eightfolds which have a non-symplectic automorphism of order $3$.
Using the theory of finite-dimensional motives, we show that the action of this
automorphism on the Chow group of $0$-cycles is as predicted by the
Bloch-Beilinson conjectures. We prove a similar statement for the
anti-symplectic involution on varieties in this family. This has interesting
consequences for the intersection product in the Chow ring of these varieties.
