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balanced n.  and  zygodrome

First numbers that belong to both sets:

11, 44, 99, 555, 666, 2222, 2233, 2266, 2277, 3300, 3366, 3377, 7755, 7799, 8811, 8822, 8855, 8877, 9900, 9944, 11111, 11155, 11444, 11555, 11666, 11888, 33399, 33555, 33777, 33999, 44000, 44400, 44422, 44433, 44488, 44555, 55500, 55522, 55566, 55577, 220000, 220033, 220088, 222000, 222266, 222888, 223322, 223333, 223388, 224466, 225555, 225566, 225577, 225588, 226600, 226611, 226644, 227744, 227766, 228866, 229911, 229922, 330022, 333000, 333666, 335544, 335566, 336666, 337700, 337722, 337744, 338811, 440000, 444466, 446655, 446666, 447744, 448866, 550011, 550077, 551133, 551155, 551199, 552222, 552277, 553333, 554444, 554466, 555111, 555222, 555444, 555566, 555577, 555888, 556600, 556611, 556644, 556677, 558811, 558844, 559933, 559966, 660033, 661166, 661177, 662222, 662233, 662299, 663322, 663333, 663366, 663377, 664411, 664444, 664477, 664499, 665500, 665522, 666222, 666600, 666655, 667711, 668877, 668899, 669944, 669999, 771111, 771155, 771177, 771188, 772266, 773344, 773388, 774411, 774422, 774444, 774488, 776655, 777222, 777744, 777999, 778822, 778833, 881166, 881177, 882200, 882244, 883344, 885533, 885555, 886666, 887700, 888333, 888999, 889933, 889955, 990000, 990088, 992200, 992299, 993333, 993355, 993366, 993388, 994444, 994466, 995500, 995544, 996611, 996644, 996688, 997700, 999666, 999966, 2200444, 2200999, 2211000, 2211100, 2211122, 2211144, 2221111, 2221166, 2221177, 2221199, 2222000, 2222111, 2222244, 2222255, 2222266, 2222444, 2223344, 2223388, 2224411, 2224444, 2224455, 2224488, 2225544, 2226600, 2226699, 2227722, 2227777, 2229999, 2233333, 2233888, 2244111, 2244422, 2244433, 2244466, 2255222, 2255333, 2255511, 2255533, 2255566, 2255577, 2255888, 2266333, 2266655, 2277755, 2277788, 2277999, 2288844, 2288866, 2299111, 2299888, 2299999, 3300044, 3300066, 3300444, 3311100, 3311166, 3311222, 3311333, 3311555, 3311888, 3322211, 3322266, 3322277, 3322666, 3331111, 3331155, 3331177, 3331188, 3332266, 3333000, 3333344, 3333388, 3333444, 3333666, 3334477, 3334499, 3335500, 3335522, 3335555, 3335588, 3336600, 3336644, 3337733, 3338822, 3338833, 3344444, 3344488, 3344888, 3344999, 3355000, 3355222, 3355333, 3355444, 3355544, 3355588, 3355888, 3366000, 3366688, 3366999, 3377000, 3377333, 3377555, 3377722, 3377766, 3377777, 3377799.

Some larger examples :