  For a multipart quantum system, a locally maximally entangled (LME) state is
one where each elementary subsystem is maximally entangled with its complement.
This paper is a sequel to arXiv:1708.01645, which gives necessary and
sufficient conditions for a system to admit LME states in terms of its
subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of
the space ${\cal H}_{LME}/K$ of LME states up to local unitary transformations
for all non-empty cases. In this paper, we provide a pedagogical overview and
physical interpretation of the the underlying mathematics that leads to these
results and give a large class of explicit constructions for LME states. In
particular, we construct all LME states for tripartite systems with subsystem
dimensions $(2,A,B)$ and give a general representation-theoretic construction
for a special class of stabilizer LME states. The latter construction provides
a common framework for many known LME states. Our results also give the
dimension of the space of SLOCC equivalence classes for states with "generic"
entanglement for all multipart systems since this space is equivalent to ${\cal
H}_{LME}/K$. Finally, we give the dimension of the stabilizer subgroup $S
\subset SL(d_1, \mathbb{C}) \times \cdots \times SL(d_n, \mathbb{C})$ for a
generic state in an arbitrary multipart system and identify all cases where
this stabilizer is trivial.
