  The three-particle Hamiltonian obtained by replacing the two-body
trigonometric potential of the Sutherland problem by a three-body one of a
similar form is shown to be exactly solvable. When written in appropriate
variables, its eigenfunctions can be expressed in terms of Jack symmetric
polynomials. The exact solvability of the problem is explained by a hidden
$sl(3,R)$ symmetry. A generalized Sutherland three-particle problem including
both two- and three-body trigonometric potentials and internal degrees of
freedom is then considered. It is analyzed in terms of three first-order
noncommuting differential-difference operators, which are constructed by
combining SUSYQM supercharges with the elements of the dihedral group~$D_6$.
Three alternative commuting operators are also introduced.
