Calogero's 'goldfish' is indeed a school of free particles

A many-body system of N nonlinear ordinary differential equations of second order which is amenable to exact treatments (a 'goldfish') (Calogero 2001 The neatest many-body problem amenable to exact treatments (a 'goldfish'?) Physica D 152-153 78-84) is shown to be equivalent through an exact transformation to the equations of one-dimensional motion of (N - 1) free particles (a school of free particles, indeed). The transformation is obtained by applying the reduction method and Lie group analysis as introduced in Nucci (1996 The complete Kepler group can be derived by Lie group analysis J. Math. Phys. 37 1772-5).