  We show that in d>1 dimensions the N-particle kinetic energy operator with
periodic boundary conditions has symmetric eigenfunctions which vanish at
particle encounters, and give a full description of these functions. In two and
three dimensions they represent common eigenstates of bosonic Hamiltonians with
any kind of contact interactions, and illustrate a partial `multi-dimensional
Bethe Ansatz' or `quantum-KAM theorem'. The lattice analogs of these functions
exist for N<=L^[d/2] where L is the linear size of the box, and are common
eigenstates of Bose-Hubbard Hamiltonians and spin-1/2 XXZ Heisenberg models.
