  We construct an integrable colored five-vertex model whose partition function
is a Lascoux atom based on the five-vertex model of Motegi and Sakai
[arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first
author, Bump, and Gustafsson [arXiv:1902.01795]. We then modify this model in
two different ways to construct a Lascoux polynomial, yielding the first known
combinatorial interpretation of a Lascoux polynomial and atom. Using this, we
prove a conjectured combinatorial interpretation in terms of set-valued
tableaux of a Lascoux polynomial and atom due to Pechenik and the second author
[arXiv:1904.09674]. We also prove the combinatorial interpretation of the
Lascoux atom using set-valued skyline tableaux of Monical [arXiv:1611.08777].
