  We study special elements of eight types (namely, neutral, standard,
costandard, distributive, codistributive, modular, lower-modular and
upper-modular elements) in the lattice EPI of all epigroup varieties. Neutral,
standard, costandard, distributive and lower-modular elements are completely
determined. A strong necessary condition and a sufficient condition for modular
elements are found. Modular elements are completely classified within the class
of commutative varieties, while codistributive and upper-modular elements are
completely determined within the wider class of strongly permutative varieties.
It is verified that an element of EPI is costandard if and only if it is
neutral; is standard if and only if it is distributive; is modular whenever it
is lower-modular; is neutral if and only if it is lower-modular and
upper-modular simultaneously. We found also an application of results
concerning neutral and lower-modular elements of EPI for studying of definable
sets of epigroup varieties.
