  The $He^+$ ion on a plane subject to a constant magnetic field $B$
perpendicular to the plane is considered taking into account the finite nuclear
mass. Factorization of eigenfunctions permits to reduce the four-dimensional
problem to three-dimensional one. The ground state energy of the composite
system is calculated in a wide range of magnetic fields from $B=0.01$ up to
$B=100$ a.u. and center-of-mass Pseudomomentum $K$ from $0$ to $1000$ a.u.
using a variational approach. The accuracy of calculations for $B = 0.1 $ a.u.
is cross-checked in Lagrange-mesh method and not less than five significant
figures are reproduced in energy. Similarly to the case of moving neutral
system on the plane a phenomenon of a sharp change of energy behavior as a
function of $K$ for a certain critical $K_c$ but a fixed magnetic field occurs.
