  The moving neutral system of two Coulomb charges on a plane subject to a
constant magnetic field $B$ perpendicular to the plane is considered. It is
shown that the composite system of finite total mass is bound for any
center-of-mass momentum $P$ and magnetic field strength; the energy of the
ground state is calculated accurately using a variational approach. Their
accuracy is cross-checked in a Lagrange-mesh method for $B=1$ a.u. and in a
perturbation theory at small $B$ and $P$. The constructed trial function has
the property of being a uniform approximation of the exact eigenfunction. For a
Hydrogen atom and a Positronium a double perturbation theory in $B$ and $P$ is
developed and the first corrections are found algebraically. A phenomenon of a
sharp change of energy behavior for a certain center-of-mass momentum and a
fixed magnetic field is indicated.
