U = 
      [[.352, .936], [.936, -.352]]

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

% Eigenvalues: 7812, -7813
U*diag([[7812, -7813]])*U' = 
      [[-5877, 5148], [5148, 5876]]

% Eigenvalues: 7813, -7812
U*diag([[7813, -7812]])*U' = 
      [[-5876, 5148], [5148, 5877]]

% Eigenvalues: 7811, -7814
U*diag([[7811, -7814]])*U' = 
      [[-5878, 5148], [5148, 5875]]

% Eigenvalues: 7814, -7811
U*diag([[7814, -7811]])*U' = 
      [[-5875, 5148], [5148, 5878]]

% Eigenvalues: 7810, -7815
U*diag([[7810, -7815]])*U' = 
      [[-5879, 5148], [5148, 5874]]

% Eigenvalues: 7815, -7810
U*diag([[7815, -7810]])*U' = 
      [[-5874, 5148], [5148, 5879]]

% Eigenvalues: 7809, -7816
U*diag([[7809, -7816]])*U' = 
      [[-5880, 5148], [5148, 5873]]

% Eigenvalues: 7816, -7809
U*diag([[7816, -7809]])*U' = 
      [[-5873, 5148], [5148, 5880]]

% Eigenvalues: 15621, -4
U*diag([[15621, -4]])*U' = 
      [[1932, 5148], [5148, 13685]]

% Eigenvalues: 15622, -3
U*diag([[15622, -3]])*U' = 
      [[1933, 5148], [5148, 13686]]

% Eigenvalues: 15625, -15625
U*diag([[15625, -15625]])*U' = 
      [[-11753, 10296], [10296, 11753]]

% Eigenvalues: 15624, -15626
U*diag([[15624, -15626]])*U' = 
      [[-11754, 10296], [10296, 11752]]

% Eigenvalues: 15626, -15624
U*diag([[15626, -15624]])*U' = 
      [[-11752, 10296], [10296, 11754]]

% Eigenvalues: 15623, -15627
U*diag([[15623, -15627]])*U' = 
      [[-11755, 10296], [10296, 11751]]

% Eigenvalues: 15627, -15623
U*diag([[15627, -15623]])*U' = 
      [[-11751, 10296], [10296, 11755]]

% Eigenvalues: 15622, -15628
U*diag([[15622, -15628]])*U' = 
      [[-11756, 10296], [10296, 11750]]

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

% Eigenvalues: 1562, -1563
U*diag([[15620, -15630]])*U'/10 = 
      [[-1175.8, 1029.6], [1029.6, 1174.8]]

% Eigenvalues: 1563, -1562
U*diag([[15630, -15620]])*U'/10 = 
      [[-1174.8, 1029.6], [1029.6, 1175.8]]

% Eigenvalues: 1561, -1564
U*diag([[15610, -15640]])*U'/10 = 
      [[-1176.8, 1029.6], [1029.6, 1173.8]]

% Eigenvalues: 1564, -1561
U*diag([[15640, -15610]])*U'/10 = 
      [[-1173.8, 1029.6], [1029.6, 1176.8]]

% Eigenvalues: 1560, -1565
U*diag([[15600, -15650]])*U'/10 = 
      [[-1177.8, 1029.6], [1029.6, 1172.8]]

% Eigenvalues: 1565, -1560
U*diag([[15650, -15600]])*U'/10 = 
      [[-1172.8, 1029.6], [1029.6, 1177.8]]

% Eigenvalues: 1559, -1566
U*diag([[15590, -15660]])*U'/10 = 
      [[-1178.8, 1029.6], [1029.6, 1171.8]]

% Eigenvalues: 1566, -1559
U*diag([[15660, -15590]])*U'/10 = 
      [[-1171.8, 1029.6], [1029.6, 1178.8]]

% Eigenvalues: 3121, -4
U*diag([[31210, -40]])*U'/10 = 
      [[383.2, 1029.6], [1029.6, 2733.8]]

% Eigenvalues: 3122, -3
U*diag([[31220, -30]])*U'/10 = 
      [[384.2, 1029.6], [1029.6, 2734.8]]

% Eigenvalues: 3123, -2
U*diag([[31230, -20]])*U'/10 = 
      [[385.2, 1029.6], [1029.6, 2735.8]]

% Eigenvalues: 3124, -1
U*diag([[31240, -10]])*U'/10 = 
      [[386.2, 1029.6], [1029.6, 2736.8]]

% Eigenvalues: 3126, 1
U*diag([[31260, 10]])*U'/10 = 
      [[388.2, 1029.6], [1029.6, 2738.8]]

% Eigenvalues: 3127, 2
U*diag([[31270, 20]])*U'/10 = 
      [[389.2, 1029.6], [1029.6, 2739.8]]

% Eigenvalues: 3128, 3
U*diag([[31280, 30]])*U'/10 = 
      [[390.2, 1029.6], [1029.6, 2740.8]]

% Eigenvalues: 3129, 4
U*diag([[31290, 40]])*U'/10 = 
      [[391.2, 1029.6], [1029.6, 2741.8]]

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

% Eigenvalues: 312, -313
U*diag([[31200, -31300]])*U'/100 = 
      [[-235.56, 205.92], [205.92, 234.56]]

% Eigenvalues: 313, -312
U*diag([[31300, -31200]])*U'/100 = 
      [[-234.56, 205.92], [205.92, 235.56]]

% Eigenvalues: 311, -314
U*diag([[31100, -31400]])*U'/100 = 
      [[-236.56, 205.92], [205.92, 233.56]]

% Eigenvalues: 314, -311
U*diag([[31400, -31100]])*U'/100 = 
      [[-233.56, 205.92], [205.92, 236.56]]

% Eigenvalues: 310, -315
U*diag([[31000, -31500]])*U'/100 = 
      [[-237.56, 205.92], [205.92, 232.56]]

% Eigenvalues: 315, -310
U*diag([[31500, -31000]])*U'/100 = 
      [[-232.56, 205.92], [205.92, 237.56]]

% Eigenvalues: 309, -316
U*diag([[30900, -31600]])*U'/100 = 
      [[-238.56, 205.92], [205.92, 231.56]]

% Eigenvalues: 316, -309
U*diag([[31600, -30900]])*U'/100 = 
      [[-231.56, 205.92], [205.92, 238.56]]

% Eigenvalues: 621, -4
U*diag([[62100, -400]])*U'/100 = 
      [[73.44, 205.92], [205.92, 543.56]]

% Eigenvalues: 622, -3
U*diag([[62200, -300]])*U'/100 = 
      [[74.44, 205.92], [205.92, 544.56]]

% Eigenvalues: 623, -2
U*diag([[62300, -200]])*U'/100 = 
      [[75.44, 205.92], [205.92, 545.56]]

% Eigenvalues: 624, -1
U*diag([[62400, -100]])*U'/100 = 
      [[76.44, 205.92], [205.92, 546.56]]

% Eigenvalues: 626, 1
U*diag([[62600, 100]])*U'/100 = 
      [[78.44, 205.92], [205.92, 548.56]]

% Eigenvalues: 627, 2
U*diag([[62700, 200]])*U'/100 = 
      [[79.44, 205.92], [205.92, 549.56]]

% Eigenvalues: 628, 3
U*diag([[62800, 300]])*U'/100 = 
      [[80.44, 205.92], [205.92, 550.56]]

% Eigenvalues: 629, 4
U*diag([[62900, 400]])*U'/100 = 
      [[81.44, 205.92], [205.92, 551.56]]