U = 
      [[3/7, 6/7, 2/7], [6/7, -2/7, -3/7], [2/7, -3/7, 6/7]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Integer matrices         %
% with integer eigenvalues %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 2, 9, -12
U*diag([[2, 9, -12]])*U' = 
      [[6, 0, -6], [0, 0, 6], [-6, 6, -7]]

% Eigenvalues: 12, -2, -9
U*diag([[12, -2, -9]])*U' = 
      [[0, 6, 0], [6, 7, 6], [0, 6, -6]]

% Eigenvalues: 3, 10, -11
U*diag([[3, 10, -11]])*U' = 
      [[7, 0, -6], [0, 1, 6], [-6, 6, -6]]

% Eigenvalues: 1, 8, -13
U*diag([[1, 8, -13]])*U' = 
      [[5, 0, -6], [0, -1, 6], [-6, 6, -8]]

% Eigenvalues: 13, -1, -8
U*diag([[13, -1, -8]])*U' = 
      [[1, 6, 0], [6, 8, 6], [0, 6, -5]]

% Eigenvalues: 4, 11, -10
U*diag([[4, 11, -10]])*U' = 
      [[8, 0, -6], [0, 2, 6], [-6, 6, -5]]

% Eigenvalues: 5, 12, -9
U*diag([[5, 12, -9]])*U' = 
      [[9, 0, -6], [0, 3, 6], [-6, 6, -4]]

% Eigenvalues: 15, 1, -6
U*diag([[15, 1, -6]])*U' = 
      [[3, 6, 0], [6, 10, 6], [0, 6, -3]]

% Eigenvalues: 6, -15, -1
U*diag([[6, -15, -1]])*U' = 
      [[-10, 6, 6], [6, 3, 0], [6, 0, -3]]

% Eigenvalues: 6, 13, -8
U*diag([[6, 13, -8]])*U' = 
      [[10, 0, -6], [0, 4, 6], [-6, 6, -3]]

% Eigenvalues: 16, 2, -5
U*diag([[16, 2, -5]])*U' = 
      [[4, 6, 0], [6, 11, 6], [0, 6, -2]]

% Eigenvalues: 5, -16, -2
U*diag([[5, -16, -2]])*U' = 
      [[-11, 6, 6], [6, 2, 0], [6, 0, -4]]

% Eigenvalues: 17, 3, -4
U*diag([[17, 3, -4]])*U' = 
      [[5, 6, 0], [6, 12, 6], [0, 6, -1]]

% Eigenvalues: 4, -17, -3
U*diag([[4, -17, -3]])*U' = 
      [[-12, 6, 6], [6, 1, 0], [6, 0, -5]]

% Eigenvalues: 8, 15, -6
U*diag([[8, 15, -6]])*U' = 
      [[12, 0, -6], [0, 6, 6], [-6, 6, -1]]

% Eigenvalues: 18, 4, -3
U*diag([[18, 4, -3]])*U' = 
      [[6, 6, 0], [6, 13, 6], [0, 6, 0]]