  We find the first non-octahedral balanced 2-neighborly 3-sphere and the
balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each
construction has 16 vertices. We show that there exists a balanced 3-neighborly
non-spherical 5-manifold with 18 vertices. We also show that the rank-selected
subcomplexes of a balanced simplicial sphere do not necessarily have an ear
decomposition.
