  The quantum mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and
$(e_2, m_2)$ subject to a constant magnetic field $B$ perpendicular to the
plane is considered. Four integrals of motion are explicitly indicated. It is
shown that for two physically-important particular cases, namely that of two
particles of equal Larmor frequencies, ${e_c} \propto
\frac{e_1}{m_1}-\frac{e_2}{m_2}=0$ (e.g. two electrons) and one of a neutral
system (e.g. the electron - positron pair, Hydrogen atom) at rest (the
center-of-mass momentum is zero) some outstanding properties occur. They are
the most visible in double polar coordinates in CMS $(R, \phi)$ and relative
$(\rho, \varphi)$ coordinate systems: (i) eigenfunctions are factorizable, all
factors except one with the explicit $\rho$-dependence are found analytically,
they have definite relative angular momentum, (ii) dynamics in $\rho$-direction
is the same for both systems, it corresponds to a funnel-type potential and it
has hidden $sl(2)$ algebra; at some discrete values of dimensionless magnetic
fields $b \leq 1$, (iii) particular integral(s) occur, (iv) the hidden $sl(2)$
algebra emerges in finite-dimensional representation, thus, the system becomes
{\it quasi-exactly-solvable} and (v) a finite number of polynomial
eigenfunctions in $\rho$ appear. Nine families of eigenfunctions are presented
explicitly.
