  Finite connected cubic symmetric graphs of girth 6 have been classified by K.
Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an
abelian automorphism group with two orbits on the vertex set. In this paper all
cubic symmetric graphs with the latter property are determined. In particular,
with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and
GP(24,5), all the obtained graphs are of girth 6.
