  A simple general theorem is used as a tool that generates nonlocal constants
of motion for Lagrangian systems. We review some cases where the constants that
we find are useful in the study of the systems: the homogeneous potentials of
degree~$-2$, the mechanical systems with viscous fluid resistance and the
conservative and dissipative Maxwell-Bloch equations of laser dynamics. We also
prove a new result on explosion in the past for mechanical system with
hydraulic (quadratic) fluid resistance and bounded potential.
