Abstract A new class of SO (3)-invariant quantum-mechanical systems, J k , of (2 k + 1) Majorana fermions is displayed. Their Fock-space description is developed. It provides, for each fixed integer k , a finite number of energy levels E J = 1 2 J(J + 1) of degeneracy (2 J + 1). The models have several striking features. One of these is the set of allowed J -values: for k = 1,2,3,…, these are J = 1 2 ; J = 3 2 ; J = 3 and J = 0 ….