Isometry groups of $4$-dimensional nilpotent Lie groups

The main purpose of this paper is to give a complete description of isometry groups on the $4$-dimensional simply connected nilpotent Lie groups. We distinguish between two geometrically distinct cases of degenerate and nondegenerate center of the group. Since Walker metrics appear as the underlying structure of neutral signature metrics on the nilpotent Lie groups with degenerate center, we find necessary and sufficient condition for them to locally admit the nilpotent group of isometries.