  We study the exact solutions of a particular class of $N$ confined particles
of equal mass, with $N=3^k \ (k=2,3,...),$ in the $D=1$ dimensional space. The
particles are clustered in clusters of 3 particles. The interactions involve a
confining mean field, two-body Calogero type of potentials inside the cluster,
interactions between the centres of mass of the clusters and finally a
non-translationally invariant $N$-body potential. The case of 9 particles is
exactly solved, in a first step, by providing the full eigensolutions and
eigenenergies. Extending this procedure, the general case of $N$ particles
($N=3^k, \ k \geq 2$) is studied in a second step. The exact solutions are
obtained via appropriate coordinate transformations and separation of
variables. The eigenwave functions and the corresponding energy spectrum are
provided.
