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Low-lying bound states for the problem of two Coulomb charges of finitemasses on a plane subject to a constant magnetic field $B$ perpendicular to theplane are considered. Major emphasis is given to two systems: two charges withthe equal charge-to-mass ratio (quasi-equal charges) and neutral systems withconcrete results for the Hydrogen atom and two electrons (quantum dot). It isshown that for these two cases, but when a neutral system is at rest (thecenter-of-mass momentum is zero), some outstanding properties occur: in doublepolar coordinates in CMS $(R, \phi)$ and relative $(\rho, \varphi)$ coordinatesystems (i) the eigenfunctions are factorizable, all factors except for$\rho$-dependent are found analytically, they have definite relative angularmomentum, (ii) dynamics in $\rho$-direction is the same for both systems beingdescribed by a funnel-type potential; (iii) at some discrete values ofdimensionless magnetic fields $b \leq 1$ the system becomes {\itquasi-exactly-solvable} and a finite number of eigenfunctions in $\rho$ arepolynomials. The variational method is employed. Trial functions are based oncombining for the phase of a wavefunction (a) the WKB expansion at largedistances, (b) the perturbation theory at small distances (c) with a form ofthe known analytically (quasi-exactly-solvable) eigenfunctions. For the loweststates with relative magnetic quantum numbers $s=0,1,2$ this approximationgives not less than 7 s.d., 8 s.d., 9 s.d., respectively, for the total energy$E(B)$ for magnetic fields $0.049\, \text{a.u.} < B < 2000\, \text{a.u.} $(Hydrogen atom) and $0.025\, \text{a.u.}\eqslantless B \eqslantless 1000\,\text{a.u.} $ (two electrons).

Two charges on plane in a magnetic field I. "Quasi-equal" charges and neutral quantum system at rest cases