  In the study of NIL-affine actions on nilpotent Lie groups we introduced so
called LR-structures on Lie algebras. The aim of this paper is to consider the
existence question of LR-structures, and to start a structure theory of
LR-algebras. We show that any Lie algebra admitting an LR-structure is 2-step
solvable. Conversely we find several classes of 2-step solvable Lie algebras
admitting an LR-structure, but also classes not admitting such a structure. We
study also ideals in LR-algebras, and classify low-dimensional real
LR-algebras.
