  We characterize 1-complemented subspaces of finite codimension in strictly
monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to
isometric isomorphism, all possible types of 1-unconditional structures in
sequence spaces with few surjective isometries. We also give a new example of a
class of real sequence spaces with few surjective isometries.
