In this toy matrix-mechanics model, the Hamiltonian has two discrete energy levels and does not commute with a second observable.
H = [[1,0],[0,2]]
X = [[0,1],[1,0]]
HX = [[0,1],[2,0]]
XH = [[0,2],[1,0]]
[H,X] = [[0,-1],[1,0]]
trace/determinant match energy levels: yes
X^2 = I : yes
[H,X] != 0 : yes