  The goal of this paper is the study of algebraic relations on the Lie algebra
of first integrals of the geodesic flow on nilpotent Lie groups equipped with a
left-invariant metric. It is proved that the isometry algebra of the $k$-step
nilpotent Lie group, $k=2,3$, gives rise to a isomorphic family of first
integrals for the geodesic flow. Also invariant first integrals are analyzed
and new involution conditions are shown. Finally it is proved that in low
dimensions complete families of first integrals can be constructed with Killing
vector fields and symmetric Killing 2-tensor fields. This holds for k-step
nilpotent Lie algebras of dimension $m\leq 5$ and $k=2,3$. The situation in
dimension six is also studied.
