U = 
      [[.28, .96], [.96, -.28]]

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

% Eigenvalues: 312, -313
U*diag([[312, -313]])*U' = 
      [[-264, 168], [168, 263]]

% Eigenvalues: 313, -312
U*diag([[313, -312]])*U' = 
      [[-263, 168], [168, 264]]

% Eigenvalues: 311, -314
U*diag([[311, -314]])*U' = 
      [[-265, 168], [168, 262]]

% Eigenvalues: 314, -311
U*diag([[314, -311]])*U' = 
      [[-262, 168], [168, 265]]

% Eigenvalues: 310, -315
U*diag([[310, -315]])*U' = 
      [[-266, 168], [168, 261]]

% Eigenvalues: 315, -310
U*diag([[315, -310]])*U' = 
      [[-261, 168], [168, 266]]

% Eigenvalues: 309, -316
U*diag([[309, -316]])*U' = 
      [[-267, 168], [168, 260]]

% Eigenvalues: 316, -309
U*diag([[316, -309]])*U' = 
      [[-260, 168], [168, 267]]

% Eigenvalues: 621, -4
U*diag([[621, -4]])*U' = 
      [[45, 168], [168, 572]]

% Eigenvalues: 622, -3
U*diag([[622, -3]])*U' = 
      [[46, 168], [168, 573]]

% Eigenvalues: 625, -625
U*diag([[625, -625]])*U' = 
      [[-527, 336], [336, 527]]

% Eigenvalues: 624, -626
U*diag([[624, -626]])*U' = 
      [[-528, 336], [336, 526]]

% Eigenvalues: 626, -624
U*diag([[626, -624]])*U' = 
      [[-526, 336], [336, 528]]

% Eigenvalues: 623, -627
U*diag([[623, -627]])*U' = 
      [[-529, 336], [336, 525]]

% Eigenvalues: 627, -623
U*diag([[627, -623]])*U' = 
      [[-525, 336], [336, 529]]

% Eigenvalues: 622, -628
U*diag([[622, -628]])*U' = 
      [[-530, 336], [336, 524]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 1 digit %
% beyond the decimal point      %
% with integer eigenvalues      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 62, -63
U*diag([[620, -630]])*U'/10 = 
      [[-53.2, 33.6], [33.6, 52.2]]

% Eigenvalues: 63, -62
U*diag([[630, -620]])*U'/10 = 
      [[-52.2, 33.6], [33.6, 53.2]]

% Eigenvalues: 61, -64
U*diag([[610, -640]])*U'/10 = 
      [[-54.2, 33.6], [33.6, 51.2]]

% Eigenvalues: 64, -61
U*diag([[640, -610]])*U'/10 = 
      [[-51.2, 33.6], [33.6, 54.2]]

% Eigenvalues: 60, -65
U*diag([[600, -650]])*U'/10 = 
      [[-55.2, 33.6], [33.6, 50.2]]

% Eigenvalues: 65, -60
U*diag([[650, -600]])*U'/10 = 
      [[-50.2, 33.6], [33.6, 55.2]]

% Eigenvalues: 59, -66
U*diag([[590, -660]])*U'/10 = 
      [[-56.2, 33.6], [33.6, 49.2]]

% Eigenvalues: 66, -59
U*diag([[660, -590]])*U'/10 = 
      [[-49.2, 33.6], [33.6, 56.2]]

% Eigenvalues: 121, -4
U*diag([[1210, -40]])*U'/10 = 
      [[5.8, 33.6], [33.6, 111.2]]

% Eigenvalues: 122, -3
U*diag([[1220, -30]])*U'/10 = 
      [[6.8, 33.6], [33.6, 112.2]]

% Eigenvalues: 123, -2
U*diag([[1230, -20]])*U'/10 = 
      [[7.8, 33.6], [33.6, 113.2]]

% Eigenvalues: 124, -1
U*diag([[1240, -10]])*U'/10 = 
      [[8.8, 33.6], [33.6, 114.2]]

% Eigenvalues: 126, 1
U*diag([[1260, 10]])*U'/10 = 
      [[10.8, 33.6], [33.6, 116.2]]

% Eigenvalues: 127, 2
U*diag([[1270, 20]])*U'/10 = 
      [[11.8, 33.6], [33.6, 117.2]]

% Eigenvalues: 128, 3
U*diag([[1280, 30]])*U'/10 = 
      [[12.8, 33.6], [33.6, 118.2]]

% Eigenvalues: 129, 4
U*diag([[1290, 40]])*U'/10 = 
      [[13.8, 33.6], [33.6, 119.2]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 2 digits %
% beyond the decimal point       %
% with integer eigenvalues       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 12, -13
U*diag([[1200, -1300]])*U'/100 = 
      [[-11.04, 6.72], [6.72, 10.04]]

% Eigenvalues: 13, -12
U*diag([[1300, -1200]])*U'/100 = 
      [[-10.04, 6.72], [6.72, 11.04]]

% Eigenvalues: 11, -14
U*diag([[1100, -1400]])*U'/100 = 
      [[-12.04, 6.72], [6.72, 9.04]]

% Eigenvalues: 14, -11
U*diag([[1400, -1100]])*U'/100 = 
      [[-9.04, 6.72], [6.72, 12.04]]

% Eigenvalues: 10, -15
U*diag([[1000, -1500]])*U'/100 = 
      [[-13.04, 6.72], [6.72, 8.04]]

% Eigenvalues: 15, -10
U*diag([[1500, -1000]])*U'/100 = 
      [[-8.04, 6.72], [6.72, 13.04]]

% Eigenvalues: 9, -16
U*diag([[900, -1600]])*U'/100 = 
      [[-14.04, 6.72], [6.72, 7.04]]

% Eigenvalues: 16, -9
U*diag([[1600, -900]])*U'/100 = 
      [[-7.04, 6.72], [6.72, 14.04]]

% Eigenvalues: 21, -4
U*diag([[2100, -400]])*U'/100 = 
      [[-2.04, 6.72], [6.72, 19.04]]

% Eigenvalues: 22, -3
U*diag([[2200, -300]])*U'/100 = 
      [[-1.04, 6.72], [6.72, 20.04]]

% Eigenvalues: 23, -2
U*diag([[2300, -200]])*U'/100 = 
      [[-.04, 6.72], [6.72, 21.04]]

% Eigenvalues: 24, -1
U*diag([[2400, -100]])*U'/100 = 
      [[.96, 6.72], [6.72, 22.04]]

% Eigenvalues: 26, 1
U*diag([[2600, 100]])*U'/100 = 
      [[2.96, 6.72], [6.72, 24.04]]

% Eigenvalues: 27, 2
U*diag([[2700, 200]])*U'/100 = 
      [[3.96, 6.72], [6.72, 25.04]]

% Eigenvalues: 28, 3
U*diag([[2800, 300]])*U'/100 = 
      [[4.96, 6.72], [6.72, 26.04]]

% Eigenvalues: 29, 4
U*diag([[2900, 400]])*U'/100 = 
      [[5.96, 6.72], [6.72, 27.04]]