A Hookean model of a three-body problem for particles with arbitrary masses and charges where two of them interact with each other through a Coulomb potential and with the third through a harmonic potential is presented. It is shown that a condition relating the masses to the harmonic coupling constants must be satisfied in order to render this problem separable. A general exact analytic solution written in terms of the relative interparticle coordinates is given as well as general expressions for the total and binding energies of this three-body system. We apply these results to examine electronic, muonic, antiprotonic, and pionic families of non-Born-Oppenheimer Hookean systems. The first contains the atoms or atomic ions: Ps- (e+ e- e-), H- (p+ e- e-), D- (d+ e- e-), T- (p+ e- e-), He4 (h e+2 e- e-), and the following molecular ions: Ps2+ (e- e+ e+), H2+ (e- p+ p+), H D+ (e- d+ p+), HT+ (e- t+ p+), DT+ (e- d+ t+), D2+ (e- d+ d+), T2+ (e- t+ t+). The muonic and antiprotonic families are similar to the electronic ones except that the species are formed replacing e- by μ- or p-. The pionic family comprises exotic atoms containing at least one pion. We also apply these results to two-electron three-dimensional spherical quantum dots and for these systems we examine the effect of electronic correlation, particularly on the singlet-triplet transitions and on the collective motion of the electrons and center of mass leading to "floppy"dynamics.