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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 anabelian automorphism group with two orbits on the vertex set. In this paper allcubic 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 andGP(24,5), all the obtained graphs are of girth 6.

Cubic symmetric graphs having an abelian automorphism group with two orbits