  We study the coupled small-amplitude motion of the mechanical system
consisting of infinitely deep water and a structure immersed in it. The former
is bounded above by a free surface, whereas the latter is formed by an
arbitrary finite number of surface-piercing bodies floating freely. The
mathematical model of time-harmonic motion is a spectral problem in which the
frequency of oscillations serves as the spectral parameter. It is proved that
there exist axisymmetric structures consisting of $N \geq 2$ bodies; every
structure has the following properties: (i) a time-harmonic wave mode is
trapped by it; (ii) some of its bodies (may be none) are motionless, whereas
the rest of the bodies (may be none) are heaving at the same frequency as
water. The construction of these structures is based on a generalization of the
semi-inverse procedure applied earlier for obtaining trapping bodies that are
motionless although float freely.
