  The one-electron Li-containing Coulomb systems of atomic type $(li, e)$ and
molecular type $(li, li, e)$, $(li, \alpha, e)$ and $(li, p, e)$ are studied in
the presence of a strong magnetic field $B \leq 10^{7}$ a.u. in the
non-relativistic framework. They are considered at the Born-Oppenheimer
approximation of zero order (infinitely massive centers) within the parallel
configuration (molecular axis parallel to the magnetic field). The variational
and Lagrange-mesh methods are employed in complement to each other. It is
demonstrated that the molecular systems ${\rm LiH}^{3+}$, ${\rm LiHe}^{4+}$ and
${\rm Li}_{2}^{5+}$ can exist for sufficiently strong magnetic fields $B
\gtrsim 10^{4}$ a.u. and that ${\rm Li}_{2}^{5+}$ can even be stable at
magnetic fields typical of magnetars.
