  As a straightforward generalization and extension of our previous paper, J.
Phys. A50 (2017) 215201 we study aspects of the quantum and classical dynamics
of a $3$-body system with equal masses, each body with $d$ degrees of freedom,
with interaction depending only on mutual (relative) distances. The study is
restricted to solutions in the space of relative motion which are functions of
mutual (relative) distances only. It is shown that the ground state (and some
other states) in the quantum case and the planar trajectories (which are in the
interaction plane) in the classical case are of this type. It corresponds to a
three-dimensional quantum particle moving in a curved space with special
$d$-dimension-independent metric in a certain $d$-dependent singular potential,
while at $d=1$ it elegantly degenerates to a two-dimensional particle moving in
flat space. It admits a description in terms of pure geometrical
characteristics of the interaction triangle which is defined by the three
relative distances. The kinetic energy of the system is $d$-independent, it has
a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden
algebra $h^{(3)}$ typical for the $H_3$ Calogero model as in the $d=3$ case. We
find an exactly-solvable three-body $S^3$-permutationally invariant,
generalized harmonic oscillator-type potential as well as a
quasi-exactly-solvable three-body sextic polynomial type potential with
singular terms. For both models an extra first order integral exists. It is
shown that a straightforward generalization of the 3-body (rational) Calogero
model to $d>1$ leads to two primitive quasi-exactly-solvable problems. The
extension to the case of non-equal masses is straightforward and is briefly
discussed.
