  The paper contains three main results. First, we show that if a commutative
semigroup variety is a modular element of the lattice Com of all commutative
semigroup varieties then it is either the variety COM of all commutative
semigroups or a nil-variety or the join of a nil-variety with the variety of
semilattices. Second, we prove that if a commutative nil-variety is a modular
element of Com then it may be given within COM by 0-reduced and substitutive
identities only. Third, we completely classify all lower-modular elements of
Com. As a corollary, we prove that an element of Com is modular whenever it is
lower-modular. All these results are precise analogues of results concerning
modular and lower-modular elements of the lattice of all semigroup varieties
obtained earlier by Jezek, McKenzie, Vernikov, and the author. As an
application of a technique developed in this paper, we provide new proofs of
the `prototypes' of the first and the third our results.

Group Theory