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balanced n.  and  palindromic n.

First numbers that belong to both sets:

2, 9, 11, 44, 99, 141, 202, 212, 232, 424, 434, 535, 555, 565, 595, 626, 666, 676, 696, 707, 737, 787, 818, 838, 868, 898, 929, 2222, 2662, 2772, 2882, 2992, 7117, 8558, 8668, 9339, 9449, 9669, 9779, 10301, 10401, 10601, 10901, 11011, 11111, 11411, 11611, 11711, 11811, 12321, 12521, 12621, 12721, 13731, 13831, 14141, 14441, 14641, 15151, 15451, 16061, 16461, 17671, 17771, 18681, 25452, 26562, 28482, 28782, 29292, 29592, 34643, 35053, 35253, 35753, 37773, 38183, 38283, 38483, 39093, 39393, 39593, 39693, 39993, 40304, 40504, 41714, 41814, 42424, 42724, 43434, 43634, 44244, 44344, 44644, 44744, 44944, 45454, 45654, 46264, 46364, 47274, 47374, 48184, 48784, 49594, 49894, 50005, 50405, 51315, 52325, 52425, 52625, 53335, 53835, 54945, 56465, 56865, 57375, 57675, 58185, 58785, 60706, 61216, 62026, 62226, 62626, 62726, 63136, 63236, 63636, 63936, 64546, 65056, 134431, 141141, 146641, 149941, 153351, 157751, 161161, 163361, 165561, 167761, 170071, 171171, 174471, 183381, 188881, 197791, 198891, 201102, 202202, 205502, 207702, 210012, 215512, 217712, 223322, 229922, 230032, 232232, 236632, 238832, 244442, 249942, 280082, 282282, 289982, 360063, 399993, 401104, 402204, 403304, 407704, 424424, 429924, 435534, 447744, 450054, 455554, 456654, 459954, 460064, 475574, 481184, 486684, 540045, 546645, 551155, 564465, 567765, 569965, 570075, 579975, 584485, 594495, 597795, 600006, 607706, 609906, 613316, 619916, 621126, 623326, 626626, 629926, 632236, 637736, 641146, 642246, 650056, 653356, 661166, 663366, 671176, 682286, 684486, 688886, 695596, 706607, 708807, 709907, 713317, 726627, 729927, 732237, 735537, 741147, 752257, 759957, 771177, 786687, 789987, 792297, 793397, 795597, 808808, 812218, 814418, 818818, 822228, 828828, 829928, 830038, 831138, 849948, 850058, 853358, 854458, 861168, 873378, 877778, 897798, 903309, 905509, 922229, 924429, 926629, 931139, 953359, 954459, 958859, 960069, 961169, 967769, 968869, 983389, 984489, 987789, 992299, 1001001, 1017101, 1020201, 1021201, 1024201, 1027201, 1036301, 2106012, 2129212.

Some larger examples :