(* Mathematica package that generates a 9-stages, non-FSAL, RKN8(6) pair *)
RKNT869[lam4_, lam5_, lam6_, lam7_, th85_, th86_, th87_, th92_, dz9_] := 
 Module[{e, dz, dz1, dz3, dz4, dz5, dz6, dz7, dz8, dzz1, dzz3, dzz4, 
   dzz5, dzz6, dzz7, dzz8, dzz, lam, lam2, lam3, lamlam, ii, z, zz, th, th21, th31,
    th32, th41, th42, th43, , th51, th52, th53, th54, th61, th62, th63, th64, th65,
    th71, th72, th73, th74, th75, th76, th81, th82, th83, th84, th91, th93, th94, 
   th95, th96, th97, mond, vthnderl, simp1, equs, equ11, so, the, thlam, thlam2, 
   zthlam}, e = {1, 1, 1, 1, 1, 1, 1, 1, 1};
  dz = {dz1, 0, dz3, dz4, dz5, dz6, dz7, dz8, dz9};
  dzz = {dzz1, 0, dzz3, dzz4, dzz5, dzz6, dzz7, dzz8, dz9 - 3/20};
  lam = {0, lam2, lam3, lam4, lam5, lam6, lam7, 1, 1};
  lamlam = DiagonalMatrix[lam]; ii = IdentityMatrix[9];
  z = dz.(ii - lamlam);
  zz = dzz.(ii - lamlam);
  th = {{0, 0, 0, 0, 0, 0, 0, 0, 0}, {th21, 0, 0, 0, 0, 0, 0, 0, 
     0}, {th31, th32, 0, 0, 0, 0, 0, 0, 0}, {th41, th42, th43, 0, 0, 0, 0, 
     0, 0}, {th51, th52, th53, th54, 0, 0, 0, 0, 0}, {th61, th62, th63, th64, 
     th65, 0, 0, 0, 0}, {th71, th72, th73, th74, th75, th76, 0, 0, 0}, {th81, 
     th82, th83, th84, th85, th86, th87, 0, 0}, {th91, th92, th93, th94, th95, 
     th96, th97, 0, 0}};
  mond = {dz.e == 1, dz.lam == 1/2, dz.lam^2 == 1/3, dz.lam^3 == 1/4, 
    dz.lam^4 == 1/5, dz.lam^5 == 1/6, dz.lam^6 == 1/7};
  vthnderl = {dzz.e - 1, dzz.lam - 1/2, dzz.(lam lam) - 1/3, 
    dzz.(lam lam lam) - 1/4, dzz.(lam lam lam lam) - 1/5, dzz.(lam lam lam lam lam) - 1/6};
  simp1 = {(z.th)[[2]] == 0, (dz.th)[[2]] == 0, (dz.(lam^2*th))[[2]] == 
     0, (dzz.th)[[2]] == 0};
  equs = {dz.(lamlam - ii).(lamlam - lam7*ii).th.(lamlam - lam3*ii).(lamlam - lam4*ii).lam == 
     Integrate[(x - 1)*(x - lam7)*
       Integrate[
        Integrate[(x - lam3)*(x - lam4)*x, {x, 0, x}], {x, 0, x}], {x, 0, 
       1}], 
    dz.(lamlam - ii).th.(lamlam - lam3*ii).(lamlam - lam4*ii).(lamlam - lam5*ii).lam == 
     Integrate[(x - 1)*
       Integrate[
        Integrate[(x - lam3)*(x - lam4)*(x - lam5)*x, {x, 0, x}], {x, 0, 
         x}], {x, 0, 1}], 
    dz.(lamlam - ii).th.(lamlam - lam3*ii).(lamlam - lam4*ii).(lamlam - lam6*ii).lam == 
     Integrate[(x - 1)*
       Integrate[
        Integrate[(x - lam3)*(x - lam4)*(x - lam6)*x, {x, 0, x}], {x, 0, 
         x}], {x, 0, 1}]};
  equ11 = 
   dzz.th.(lamlam - lam3 ii).(lamlam - lam4 ii).lam - 
    lam2*(lam2 - lam3)*(lam2 - lam4)*(dzz.th)[[2]] - 
    Integrate[
     Integrate[
      Integrate[(x - lam3)*(x - lam4)*x, {x, 0, x}], {x, 0, x}], {x, 0, 
      1}];
  lam3 = (15 - 20*lam4 - 20*lam5 + 28*lam4*lam5 - 20*lam6 + 28*lam4*lam6 + 28*lam5*lam6 - 
      42*lam4*lam5*lam6 - 20*lam7 + 28*lam4*lam7 + 28*lam5*lam7 - 42*lam4*lam5*lam7 + 
      28*lam6*lam7 - 42*lam4*lam6*lam7 - 42*lam5*lam6*lam7 + 
      70*lam4*lam5*lam6*
       lam7)/(2*(10 - 14*lam4 - 14*lam5 + 21*lam4*lam5 - 14*lam6 + 21*lam4*lam6 + 
        21*lam5*lam6 - 35*lam4*lam5*lam6 - 14*lam7 + 21*lam4*lam7 + 21*lam5*lam7 - 
        35*lam4*lam5*lam7 + 21*lam6*lam7 - 35*lam4*lam6*lam7 - 35*lam5*lam6*lam7 + 
        70*lam4*lam5*lam6*lam7));
  lam2 = lam3/2;
  so = Solve[mond, {dz1, dz3, dz4, dz5, dz6, dz7, dz8}];
  {dz1, dz3, dz4, dz5, dz6, dz7, dz8} = Simplify[so[[1, All, 2]]];
  the = th.e - lam^2/2; thlam = th.lam - lam^3/6; thlam2 = th.lam^2 - lam^4/12; 
  th32 = lam3^3/6/lam2;
  so = Solve[{thlam[[4]] == 0, thlam2[[4]] == 0}, {th42, th43}]; {th42, th43} = 
   Simplify[so[[1, All, 2]]];
  so = Solve[simp1, {th82, th72, th62, th52}]; {th82, th72, th62, th52} = 
   Simplify[so[[1, All, 2]]];
  so = Solve[equs, {th65, th76, th75}]; {th65, th76, th75} = 
   Simplify[so[[1, All, 2]]];
  so = Solve[
    Join[thlam[[5 ;; 8]], thlam2[[5 ;; 8]]] == Array[0 &, 8], {th53, th54, 
     th63, th64, th73, th74, th83, th84}];
  {th53, th54, th63, th64, th73, th74, th83, th84} = Simplify[so[[1, All, 2]]];
  zthlam = dz.(th - lamlam^2/2 + lamlam - ii/2);
  so = Solve[
    zthlam[[3 ;; 7]] == Array[0 &, 5], {th93, th94, th95, th96, th97}];
  {th93, th94, th95, th96, th97} = Simplify[so[[1, All, 2]]];
  so = Solve[
    the[[2 ;; 9]] == Array[0 &, 8], {th21, th31, th41, th51, th61, th71, th81,
      th91}];
  {th21, th31, th41, th51, th61, th71, th81, th91} = so[[1, All, 2]];
  so = Solve[{Join[vthnderl, {equ11}] == {0, 0, 0, 0, 0, 0, 0}}, {dzz1,
      dzz3, dzz4, dzz5, dzz6, dzz7, dzz8}];
  {dzz1, dzz3, dzz4, dzz5, dzz6, dzz7, dzz8} = 
   Simplify[so[[1, All, 2]]]; Return[{th, lam, z, zz, dz, dzz}]];


(* the proposed pair *)
(* produced by the above algorithm by calling
RKNT869[5601632/13092959,25660393/34815795,44986679/52545954,14200983/14248358,-(187948636/42720231),361348112/36989561,-(70523021/17471878),163509818/17684341,8502977/39270418]
*)
th={{0,0,0,0,0,0,0,0,0},
{0.0025840643315442907321,0,0,0,0,0,0,0,0},
{0.0034454191087257209762,0.0068908382174514419524,0,0,0,0,0,0,0},
{0.089310053243512999788,-0.17713368063146569000,0.17934521334342683149,0,0,0,0,0,0},
{-1.3171369895961329121,3.2896224231964203548,-1.9641671350187717429,0.26329058993831411794,0,0,0,0,0},
{20.550393754319129783,-49.999977200234258985,32.021126002070149264,-2.3295512950988886116,0.12449635970173263193,0,0,0,0},
{-13.826921872061198090,33.759962680871584778,-21.282826013990816756,1.8230049969590903569,0.016655790313559262944,0.0068050010053952783074,0,0,0},
{-15.952386550468155559,30.671787049072662871,-12.383277707248677475,-3.1691479524228384264,-4.3995229332912549092,9.7689213451330228007,-4.0363732507747593018,0,0},
{-20.197045678024291101,9.2460226818743203380,27.900109367960859296,-22.825666274504877393,-21.364496155983673514,47.293139735736257071,-19.552063677058594697,0,0}};
lam={0,0.071889697892595024205,0.14377939578519004841,0.42783544957255269798,0.73703309087154264322,0.85613973247112422776,0.99667505546954954388,1.0000000000000000000,1.0000000000000000000};
z={0.041165193884034402069,0,0.19643817587185774907,0.18274409598642543708,0.070446866095352504692,0.0061068584104038427917,0.0030988097519260642956,0,0};
zz={0.041683647470295155665,0,0.19507569083352129853,0.18493492911788562189,0.065649597275375841906,0.010694085671282908655,0.0019620496316391733545,0,0};
dz={0.041165193884034402069,0,0.22942472407797259127,0.31939080435847117664,0.26789251289765793529,0.042449930862099003510,0.93198840568514519504,-1.0488352919865401601,0.21652372022115985626};
dzz={0.041683647470295155665,0,0.22783344604561794335,0.32321983069333145037,0.24964965171076527931,0.074336617434250088689,0.59009995916333654414,-0.57334687273875631779,0.066523720221159856256};