  Euclidean billiard partitions are introduced by Andrews, Dragovic and
Radnovic in their study of periodic trajectories of ellipsoidal billiards in
the Euclidean space. They are integer partitions into distinct parts such that
(E1) adjacent parts are never both odd; (E2) the smallest part is even. In this
paper, we prove bivariate generating function identities that keep track of
both the size and length not only for Euclidean billiard partitions but also
for distinct partitions satisfying merely Condition (E1). In analogy, we also
investigate distinct partitions such that adjacent parts are never both even.
