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In this work we study 2- and 3-body oscillators with quadratic and sexticpairwise potentials which depend on relative distances, $|{\bf r}_i - {\bf r}_j|$, between particles. The two-body harmonic oscillator is two-parametric andcan be reduced to a one-dimensional radial Jacobi oscillator, while in the3-body case such a reduction is not possible in general. Our study isrestricted to solutions in the space of relative motion which are functions ofmutual (relative) distances only ($S$-states). We pay special attention to thecases where the masses of the particles and spring constants are unequal aswell as to the atomic, where one mass is infinite, and molecular, where twomasses are infinite, limits. In general, three-body harmonic oscillator is7-parametric depending on 3 masses and 3 spring constants, and frequency. Inparticular, the first and second order integrals of the 3-body oscillator forunequal masses are searched: it is shown that for certain relations involvingmasses and spring constants the system becomes maximally (minimally)superintegrable in the case of two (one) relations.

Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$