Here are "all" the solutions (x,y,z) with max(|x|,|y|,|z|)<10^20 and n<200: (First colum (or two) shows which multiple of the generator(s) these are.) ("all" means I checked all small sums of generators -- to 10g in the rank-1 case and to 8g1 +- 8g2 in the rank-2 case.) Start with the rank-1 cases; rank-2 are below n=6 1 [1, 2, 3] 2 [-52, 19, 21] 3 [1817, 3258, 5275] 4 [-2847511, -124904, 3096807] 5 [4904676969, 10840875082, 15051171563] 6 [-203863624933571, -150777667094725, 458665691607396] 7 [29381282043563909553, 46875396961726681714, 81821352777652044467] 9 1 [2, 3, 7] 2 [-632, -133, 1005] 3 [970703, 2982043, 4461282] 4 [-643014345584, 168578975369, 251333747655] 5 [416909182983584127, 1813259570662629803, 1908271675841583778] 10 1 [5, 7, 18] 2 [-27445, -3924, 39949] 3 [4192875343, 11021882957, 19765145610] 4 [-1751428277352936985, 331294903543751976, 823425682888111849] 13 1 [9, 13, 38] 2 [-474075, -55784, 703859] 3 [2197345737653, 6384056084353, 12689495542854] 14 1 [2, 7, 13] 2 [-15323, 3708, 4355] 3 [279025573, 759054842, 1638591583] 4 [-13646739810262536, -484433089497049, 15890244296549545] 15 1 [-7, 1, 3] 2 [-185, -91, 516] 3 [-801173, 26147, 671769] 4 [-25556208395, 2878275864, 13078494611] 5 [-20473048106534141, 4391597856256257, 6129300601851707] 16 1 [-70, 9, 31] 2 [-3355119, -2034340, 10655599] 3 [-3527693515451449, 196284358893169, 2519051221242990] 17 1 [5, 18, 37] 2 [-909504, 211159, 224105] 3 [1932849997397, 7649960172210, 14857581287413] 18 1 [13, 42, 95] 2 [-35917476, 6829645, 10182731] 3 [5902844861231317, 35013190193908290, 54059017558123943] 19 1 [1, 5, 9] 2 [-910, 151, 279] 3 [728051, 1279935, 4135819] 4 [-211206892329, 16629966080, 117068584489] 5 [51055079453588331, 484586928083777219, 484892826532842775] 20 1 [-14, -13, 61] 2 [-3208492, 33367, 2986425] 3 [-144397966883171, -115599556803821, 584500485805418] 21 1 [2, 13, 21] 2 [-45969, -14128, 120289] 3 [38304582498, 44899033717, 187979061005] 4 [-80139359532221561649, 11345585147381763473, 25962379050245471584] 26 1 [9, 38, 91] 2 [-28607996, 4927013, 6288291] 3 [2592527851712161, 16461714780091854, 31072284713059955] 29 1 [27, 43, 182] 2 [-160624647, -10887968, 258382055] 3 [725188306504448123, 2863730199603918763, 7554216031389795222] 30 1 [2, 21, 31] 2 [-625443, 41060, 286843] 3 [907576024698, 2555537666039, 8213238158509] 31 1 [-27, -1, 37] 2 [-683829, 35168, 364117] 3 [-12626142821795, 986557671783, 4904613318557] 35 1 [-19, -14, 97] 2 [-17392923, 399155, 12873448] 3 [-5758144589944571, -2294321839711511, 21719405222496338] 36 1 [-78, -7, 151] 2 [-268576932, 27422521, 71605559] 3 [-5627055531704328233, 769604001735890105, 1130663516528917098] 38 1 [70, 151, 629] 2 [-17179066660, -1949869179, 37525793539] 40 1 [-9, 1, 2] 2 [-1460, 63, 737] 3 [-4268359, -502289, 9685062] 4 [-584095638760, -221284447839, 2293828516639] 5 [-11215515962305226511, 107397880364505418, 9792290232973328039] 44 1 [-819, 19, 554] 2 [-304345505372, 13668309737, 139250151495] 47 1 [-845, 38, 367] 2 [-41722712395, -24805715544, 221450000899] 51 1 [9, 13, 77] 2 [-1022256, -28259, 1481363] 3 [9496944543173, 28497283786885, 116604793962657] 53 1 [2, 7, 27] 2 [-7736, -1809, 27545] 3 [210121627, 5309015927, 5755076082] 4 [-161721511837088944, 12589342705868215, 38644042894056729] 54 1 [2, 43, 57] 2 [-647349, -30196, 1137565] 3 [370030298454, 3412808117911, 7948993687541] 57 1 [19, 91, 310] 2 [-301150759, 25720080, 61301239] 3 [278307036741995371, 5726573130751998070, 8251115886938879299] 62 1 [-1950953, -1513300, 13559153] 63 1 [-3775, 247, 903] 2 [-24295747136997, 1361350133800, 6734754327197] 64 1 [-232736, -119479, 1338039] 66 1 [1, 3, 14] 2 [-633, 28, 209] 3 [55075, 1201649, 1852326] 4 [-7357433048, -5764969641, 53014540601] 5 [2146353636910641, 32426975604751798, 63703866521074499] 67 1 [1133, 7525, 23517] 2 [-12232457245758050, 1248321775926537, 1781635744669913] 70 1 [-896668, -27083, 1478979] 71 1 [-67, 7, 9] 2 [-1354977, 12931, 1055222] 3 [-16533643269215, -7773271929547, 95796860400927] 72 1 [-5450170263655, 404512675962, 1012930784383] 73 1 [89200900157319, 1391526622949983, 2848691279889518] 74 1 [133, 2502, 4607] 2 [-244642267132812, 10921761369155, 72146437148197] 83 1 [5, 9, 61] 2 [-510426, 9211, 282815] 3 [406164641531, 2343744686659, 8805786469335] 84 1 [-56, 1, 31] 2 [-185360, -22823, 604903] 3 [-3781070788603, 264046351688, 641207037467] 86 1 [2, 73, 91] 2 [-35399819, -729108, 55010099] 3 [5660399432462138, 114038591571428467, 220904967896959585] 87 1 [-21, 1, 5] 2 [-23155, 1302, 4693] 3 [-2681061091, 13866001, 2282398335] 4 [-16298444136760464, -2342189493794095, 58272293965838719] 92 1 [-35661385544981, 548624531286, 20446843218005] 96 1 [-38, 3, 5] 2 [-274495, 3724, 164991] 3 [-376883047187, -80993577493, 1720995246990] 98 1 [-14154192, -2559169, 59978401] 99 1 [-14466072543, 1150522313, 1832602198] 101 1 [79, 1271, 3078] 2 [-37063297379023, 2141532398239, 6318310548816] 102 1 [459338480695732254, 3816006884967068935, 13212742329826830581] 103 1 [-9, -4, 61] 2 [-58383, 1159, 26024] 3 [-30685083497, -2619142867, 92681703468] 4 [-5178889801145049584, 251071310758255049, 1028892954900722535] 107 1 [-197624310994, -83341950251, 1329876450605] 108 1 [-7, -2, 39] 2 [-415289, 13065, 119324] 3 [-1207414568807, -21659465017, 1932665115654] 109 1 [-7254524660292, 99054267227, 4035385003297] 110 1 [1147, 2745, 18578] 2 [-17596934037664605, 356226463814956, 7330896793887269] 112 [Solutions exist but are too large for this list.] 113 1 [345842, 6313383, 15170275] 116 1 [-1204, 13, 739] 2 [-1289806157279, 27935974079, 485911791288] 117 1 [545, 1677, 10318] 2 [-596093532925955, -46992269360344, 1841855805999339] 119 1 [-49, 1, 19] 2 [-1117675, 62254, 168021] 3 [-23734964175841, 1379659674059, 3420353969221] 120 1 [-8869, 496, 1317] 2 [-347156560777312, -19177421644913, 918936911786865] 122 [Solutions exist but are too large for this list.] 123 1 [-10554611259665663, 45191178833837, 8723981310176706] 124 1 [-4389003335, 75992904, 1882858151] 127 1 [-931, 45, 151] 2 [-18233942445, -1560275003, 60931944008] 128 1 [-1158179, 23611, 422318] 130 [Solutions exist but are too large for this list.] 132 1 [-2234, 39, 905] 2 [-10090214441815, 463732094631, 1655747655604] 133 [Solutions exist but are too large for this list.] 142 [Solutions exist but are too large for this list.] 143 1 [-1636453, 2435, 1520593] 145 1 [44634584148027469, 157591646586434781, 1007950541819512850] 147 1 [21, 925, 1529] 2 [-826614601475, 14611305261, 302529417014] 148 1 [-1418519131294563, 71841303293459, 188778746384314] 155 1 [-466371, 458, 442729] 156 1 [-151742066509610694, 2433329851945933, 57378032205801587] 158 1 [5642215349875, 7336556299898, 80828288788977] 159 1 [-31, 1, 6] 2 [-178752, 6665, 30007] 3 [-191720116529, 483120431, 166286747034] 4 [-38247451831347664415, -4776759358338903936, 171394870282816662431] 162 1 [35, 1854, 2881] 2 [-44334184670964, 613899299195, 18359866789309] 164 [Solutions exist but are too large for this list.] 166 1 [9, 611, 790] 2 [-301246383581, 2384458821, 180197737580] 167 1 [-868179733296745, -90197542563908, 3641058343253213] 172 1 [-23593229783585883, 127237919517328, 16590668075015195] 175 1 [-55614086497, 1343816497, 12984427575] 177 [Solutions assumed to exist but are probably too large for this list.] 178 1 [2, 27, 97] 2 [-4928391, 357196, 381695] 3 [14392834313297, 170840056879242, 655798399654747] 181 1 [10672860536839861, 21088064331923949, 201705586625136962] 185 1 [-175, -4, 379] 2 [-9527000525, 239197256, 2031178869] 187 [Solutions exist but are too large for this list.] 189 1 [-18396, 209, 7891] 2 [-49125036148841315, 1403810784530363, 9038814254534232] 190 [Solutions exist but are too large for this list.] 191 1 [-56059, 1399, 11655] 2 [-1026656287664235795, 44299663679437192, 124339525164402923] 192 1 [-3345, 38, 1417] 2 [-9516939248145, -1530353758844, 53034545735249] 197 1 [127, 6278, 11655] 2 [-9939340825013776, 169642439406721, 2883849663571695] Here are the rank-2 cases. These lists hand-sorted, sorry. 41 1 0 [1, 2, 9] 1 -1 [1, 5, 14] 0 1 [4, 1, -13] 1 1 [2, 31, 43] 2 0 [-208, 9, 103] 2 -1 [-436, 13, 271] 1 -2 [61, 1133, 1314] 0 2 [-1256, 117, 323] 2 1 [-8305, 1261, 1324] 1 2 [1541, 10690, 25029] 2 -2 [-13715, 1736, 2619] 3 -1 [13547, 17314, 97663] 3 0 [11441, 86425, 192834] 1 -3 [240322, 681959, 2567203] 3 1 [193669, 2829857, 4119086] 3 -2 [129986, 1478133, 2573269] 0 3 [-6313051, 76535, 5278156] 2 2 [-1280669, -99432, 2464469] 2 -3 [-9291739, -1228292, 22444519] 1 3 [19870189, 27931673, 150193166] 3 -3 [49375661, 503950510, 940181369] 4 -1 [-393235219, 5949980, 313300159] 4 0 [-227135584, -90824751, 926963023] 4 -2 [-1239505272, -190885253, 3208796909] 3 2 [1189303714, 2163614573, 10214929341] 2 3 [-64230236761, 2095209733, 38043940636] 1 -4 [21889228066, 77933929545, 260937225569] 4 1 [-113677947868, 8294022721, 36707476819] 2 -4 [-367176192097, 7096282561, 272972735712] 0 4 [-640504745767, 40313397424, 235764767511] 4 -3 [-1709696670292, 201606816389, 350015854711] 5 -1 [1305907447649, 4068724293358, 14597316416261] 3 -4 [4620332860137, 5150126215361, 31113448677154] 1 4 [2313004281209, 28263232405602, 46888422929521] 5 0 [4154246163394, 80100285713657, 88525610444241] 5 -2 [10984950177181, 35643761036685, 125224315795874] 3 3 [12901838728418, 70556481743395, 188269985477767] 4 2 [-103607707443227, -374671346445, 108724253790392] 5 -3 [377031493149127, 7377510714877858, 7983955441197731] 5 1 [873867471084863, 3991394203757410, 11727180778789867] 4 -4 [-6770219962006089, 174624405217145, 4509732961975024] 2 4 [-4354874821588113, -3182787253684207, 23939153496242560] 1 -5 [9479215702762409, 191138780413831901, 197118040212610702] 2 -5 [-172519997643080009, -79468492739010091, 754701741610738300] 0 5 [-4043517017112380107, 453758477288835884, 868878445919509831] 3 -5 [431729494087189747, 3675341541489181666, 7652356808774912183] 6 -1 [-12225197308817341804, 709291932808347377, 4822765657669842259] 4 3 [-16746694868427900997, 1357287213847030804, 4909978063211579401] 5 -4 [6034953147736688226, 26492026297942565137, 79487929060807347929] 5 2 [9473776430012344981, 20826235266050325557, 89324025186302233794] 6 -2 [-16085217545504914136, 264093988932549323, 12548930324638321341] 6 0 [-88511859555119590825, 10664630689570313977, 17741691932442396048] 69 1 0 [2, 57, 73] 0 1 [42, 95, 523] 1 -1 [-6631, 676, 939] 1 1 [-171639, 10564, 39899] 2 0 [-13518505, -407648, 22173513] 2 -1 [38808119, 45866266, 349822755] 0 2 [-13583250005, 409659101, 5972328264] 1 -2 [907290117, 16844207218, 29911475693] 2 1 [11708394650, 69802887831, 234380785219] 1 2 [1127308588373, 8812509674482, 25666209109533] 2 -2 [-274070675111613, 3441623231333, 197658393760360] 3 0 [945236516322402, 13341382667720153, 28024584728207113] 3 -1 [-12868715364132036, -6605513773470859, 76790717731196839] 2 2 [-54087597069217597352, -10699546300421376645, 201795362835514017437] 76 1 0 [-45, 2, 13] 1 -1 [-6437, 266, 1989] 0 1 [-14911, 1327, 2196] 2 -1 [-1738871, 4472, 1625751] 2 0 [-186644, -98505, 1184729] 1 1 [-10308142, -452069, 20208069] 3 -1 [-240199268101, -42466863834, 889330019557] 3 0 [-18248210699597, 368607865517, 9956121489990] 1 -2 [-63263144517229, 1556049233997, 30171140545078] 2 1 [-16022586064007, -537811790841, 28289738732576] 2 -2 [-530537340215961, 50529829299401, 73039774952452] 0 2 [-7285488014219544, 123064550019983, 4413430505907409] 3 -2 [-860292781038160301, -99884565342548082, 2603010608841911117] 77 1 -1 [-52, 5, 7] 1 0 [67, 630, 1763] 0 1 [133, 1382, 3665] 2 -1 [40225, 221062, 819413] 2 -2 [-704755, -11336, 985131] 1 -2 [401247, 1986038, 7768135] 3 -2 [8123011655, 138755312182, 277792875423] 1 1 [-286888709233, -617984852, 302735049565] 2 0 [-3452022745920, 350386881449, 440302615831] 2 -3 [321489851593, 5998665668870, 11380945916077] 0 2 [-9665199943115, -6196415424381, 68031371247416] 3 -3 [-6797222524877, -2472874880783, 36102438384740] 3 -1 [-14340753509409155, 140572907970988, 10825938096471407] 1 -3 [-5877493766913240823, 77508999432132100, 3984620000322516923] 4 -3 [8621437801077187997, 24553110171777495910, 127193505400632064313] 94 1 0 [27, 182, 673] 0 1 [19, 746, 945] 1 -1 [-52056, 2569, 11111] 1 1 [-6133771, 182836, 2101059] 2 0 [-55473879188, 4043979605, 8067401523] 2 -1 [20400692347, 64738300490, 351211722633] 0 2 [-392320602765, -8146206091, 629550717436] 1 -2 [180053104598, 478460823507, 2838538679977] 2 1 [573043116672162, 709974221647523, 6176982170933905] 1 2 [1045140774118987, 17101422573331073, 39410179847609022] 2 -2 [-1567536726449591375, 70522666866122832, 365914133782922543] 105 1 0 [2, 91, 111] 0 1 [35, 1171, 1854] 1 1 [-3171, -97, 6124] 1 -1 [-5669524, 38037, 4367911] 2 1 [4934775, 86738143, 204325982] 2 0 [-83645493, -1228120, 124453693] 1 2 [22107891903, 239901074434, 733520068619] 0 2 [-2976931342944, -166847122855, 7462479698119] 2 -1 [3247719051935, 164572391923863, 175124516201662] 2 2 [-728288603318787, 25370916851995, 195259187038712] 3 1 [-68861204583395076, 1805032685147293, 23964331551703919] 1 -2 [16738475957549723, 373159048714477474, 766861901495659599] 3 0 [28437542761398498, 776625821833283479, 1409469972958614419] 106 1 0 [1, 35, 54] 1 1 [1342, 15929, 46683] 0 1 [-372209, 5697, 195944] 2 0 [-5511205, 114589, 2315196] 2 1 [-11065988, 466949, 2447091] 1 -1 [100054843, 4555645497, 5608864334] 3 1 [327256085169, 710839851638, 4957711976947] 3 0 [37576427331870, 289465199522171, 1063063406432089] 2 -1 [-159742867359858741, -1163991786521804, 200181281112233117] 1 2 [155482701049777507, 528780627994949606, 2943367501625864289] 2 2 [-188566488611747883, -131106239737391116, 1620520267959562771] 126 1 0 [2, 111, 133] 1 1 [-1371, -245, 6536] 0 1 [1093, 4199, 23982] 2 1 [843543, 6610037, 26297374] 2 0 [-181893859, -1970012, 261141819] 1 -1 [-1299069644, 70342047, 189781529] 1 2 [1437546238546, 8374810124997, 38751798984143] 2 2 [-382821621974751, 16747073099696, 69038625084415] 3 1 [-1562019291714084, 40038875821463, 470419373910289] 0 2 [-57910985383835389, 1744195061665644, 14994740462918917] 2 -1 [67734249302807647, 126717054608677245, 1038850753947890438] 3 0 [123647398756761978, 4400373539103726365, 7569639732064906567] 3 2 [2953793474304361458, 8405086108669284847, 55830914390292366821] 129 1 0 [31, 774, 1679] 0 1 [70, 629, 2361] 1 -1 [-76, -39, 619] 2 -1 [11970393, 28883125, 210898982] 1 -2 [11235206, 80300179, 338895771] 2 -2 [-18029934328, 235007683, 9267009765] 1 1 [-6763090191, -126682669, 11702817844] 2 0 [-3663450397152, 132354035465, 778476800407] 0 2 [-8278035566149, 586744361229, 903847958440] 2 -3 [75246303764889437, 2289026596335072678, 4416237184193680945] 3 -2 [20679588657162219, 1246555575627259139, 1389857356755760966] 3 -1 [-27735911629555249676, -2373138041209629229, 93378625764176926641] 136 1 0 [-203, -45, 1118] 0 1 [-7141, 126, 2797] 1 1 [-755668, 10791, 350329] 1 -1 [-33037456, 2347177, 3414231] 2 0 [-283693749871, 9250669636, 63260120655] 2 1 [-46337552518243, 3898283767654, 4044883546251] 0 2 [-156241618021777, -48639631294044, 1018525604957809] 1 2 [-12074932284209123, 41712851784123, 10201735152568634] 2 -1 [-19750544369295219, -365366133274978, 34691290356420859] 1 -2 [-19499587573556334338, 533238185622441299, 5146622717034419253] 149 1 0 [1, 14, 45] 0 1 [2, 133, 157] 1 1 [-148, -7, 403] 1 -1 [-104923, 2623, 27652] 2 1 [45257, 87913, 769298] 2 0 [-1275736, 88381, 123435] 1 2 [1261745, 32670622, 75361293] 0 2 [-369362753, -3034512, 514694705] 2 2 [-9686773160, 480848333, 1306303947] 2 -1 [3617906033, 3624015553, 44175121682] 3 1 [-174676030828, -32663933033, 925166876893] 1 -2 [155269296833, 1140528906910, 5108338064637] 3 0 [693682595369, 5078527204590, 22784790354661] 3 2 [33048307741225, 125476538605182, 784414933611893] 1 3 [-294267517701885403, 3342022860632863, 150590290924215268] 2 3 [107798095038874577, 405521326147169473, 2546914656539833202] 0 3 [476356977054008162, 21419375371536908893, 35233086477896196133] 3 -1 [-11717115935424608677, 293442660915713057, 3082780187589773788] 2 -2 [-3085234835318431325, -2216560071134270043, 31940773732903973768] 151 1 0 [-13, -9, 133] 0 1 [-1054, 9, 637] 1 -1 [-8611, 351, 1393] 1 1 [-1379209, 81385, 154539] 2 0 [-15296879, 97622, 10596753] 2 -1 [-100662308, 1475435, 42143913] 1 -2 [-374313875377, 9082811115, 100197596797] 2 1 [-1788916676508, 4562327269, 1560698646983] 0 2 [-272431726696, -12864422853, 745867244941] 2 -2 [-112531099927194, -11451777106283, 444745232014013] 1 2 [-9602601659947141, 340750186252063, 1780604238568077] 3 -1 [-8714709349340571, 243978975533243, 2035166782540801] 3 0 [-3042558612181195, -1008723050634647, 21558942425289087] 154 1 0 [2, 13, 63] 0 1 [62, 1183, 3285] 1 1 [-58289, -936, 101729] 1 -1 [-151619, 11259, 13244] 2 0 [-3250507, 137907, 495700] 2 -1 [94550101, 2427158214, 5731153295] 2 1 [394133054, 847190695, 7164362061] 0 2 [-286202540385, -110273782924, 2207162686729] 1 2 [689430032438, 33597986722807, 53457471559053] 1 -2 [2025699056437, 18110719408158, 74632920657023] 3 0 [1693932790087, 101518256992698, 137351004492437] 3 -1 [-364137791834683376, 4291680539385239, 177418745435198601] 3 1 [-109984876421351084, -9890849401424091, 413219091513078851] 2 -2 [-2430548300078448088, 7148407995585805, 2066790518282886723] 2 2 [-3229740787753213195, 61619104026400752, 1060349876636379883] 160 1 0 [-4559, 182, 711] 0 1 [-3691, 43, 1764] 1 1 [-450338, 8325, 146963] 1 -1 [-300499659, 22385734, 25186115] 0 2 [-88701589656792, 2398251904945, 20259722706767] 2 0 [-17311089124420, -1611136298417, 67376121464817] 1 2 [-289634950450809, 1806984872017, 197773288015646] 2 1 [-45758042357317, -18933901762491, 372687283289104] 2 -1 [-42636279147333150267, -778720031458682936, 79306872497525264699] 161 1 -1 [-95, 7, 8] 1 0 [11, 38, 259] 0 1 [109, 3933, 7826] 2 -1 [146, 6517, 11349] 2 -2 [-6861744, 16055, 6005209] 1 -2 [39927179, 179799907, 1072531846] 2 0 [-190510177, -13867119, 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