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.
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