The geodesic flow on nilpotent Lie groups of steps two and three

<p style='text-indent:20px;'>The goal of this paper is the study integrability geodesic flow on <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula>-step nilpotent Lie groups, = 2, 3, when equipped with a left-invariant metric. Liouville proved in low dimensions. Moreover, it shown that complete families first integrals can be constructed Killing vector fields and symmetric 2-tensor fields. This holds for dimension id="M2">\begin{document}$ m\leq 5 $\end{document}</tex-math></inline-formula>. The situation six similar most cases. Several algebraic relations algebra are explicitly written. Also invariant analyzed several involution conditions shown.</p>