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cyclic n.  and  zygodrome

First numbers that belong to both sets:

11, 33, 77, 1111, 1133, 1177, 1199, 3333, 3377, 4411, 4433, 4499, 5511, 5533, 5599, 6611, 6677, 7711, 7733, 7777, 7799, 8877, 8899, 9911, 9977, 11111, 11155, 11177, 11333, 11777, 22111, 22211, 22277, 22299, 22333, 22555, 22777, 22999, 33311, 33377, 33555, 44111, 44333, 44411, 44455, 44477, 44555, 44777, 44999, 55111, 55333, 55511, 55533, 55577, 55999, 66111, 66611, 66677, 66777, 66999, 77111, 77333, 77711, 77777, 77999, 88111, 88333, 88555, 88777, 88811, 88855, 88877, 88899, 88999, 99933, 99977, 110011, 110077, 110099, 111133, 112277, 113311, 113333, 114411, 114433, 114499, 115511, 115577, 116611, 116677, 117799, 118811, 118877, 119933, 119977, 119999, 220011, 220033, 221111, 221133, 221177, 222211, 222299, 223333, 224477, 224499, 225511, 225577, 225599, 226677, 226699, 227711, 227777, 228833, 228877, 228899, 229933, 330011, 330077, 331111, 331133, 331199, 332211, 332233, 332299, 333311, 333377, 333399, 334433, 334477, 334499, 335577, 335599, 336611, 336677, 337733, 337777, 337799, 338833, 338899, 339933, 339977, 440011, 440033, 440099, 441111, 441177, 441199, 442277, 443311, 443333, 443377, 443399, 444477, 445511, 445577, 446633, 446677, 446699, 447733, 447799, 448811, 448877, 449911, 449933, 449977, 550099, 551111, 551199, 552211, 552233, 552277, 553399, 554411, 554433, 555533, 555599, 556611, 556633, 556699, 557711, 557733, 557777, 558811, 558833, 559933, 559977, 661111, 661133, 661177, 661199, 662299, 663311, 663377, 664433, 664477, 664499, 665533, 665599, 666611, 666633, 666677, 667711, 667733, 668833, 669911, 770011, 770033, 770077, 770099, 771177, 771199, 772211, 772277, 773333, 773377, 773399, 774499, 775511, 775577, 776611, 776633, 776677, 777711, 778811, 778899, 779911, 779977, 880077, 880099, 881111, 881177, 882233, 882277, 882299, 883333, 883399, 884411, 884433, 884477, 885511, 886633, 886699, 887711, 887799, 888833, 888877, 888899, 889933, 889999, 990011, 990077, 991133, 991177, 991199, 992211, 992233, 993311, 994411, 994433, 994477, 995599, 996699, 997711, 997733, 997777, 997799, 998833, 998877, 999911, 999977, 1100033, 1100077, 1100111, 1100555, 1100777, 1100999, 1110011, 1110077, 1111111.

Some larger examples :