RKO5[c1_, c2_, c5_] :=
Module[{b, b2, b3, b4, b5, c, a, a32, a42, a43, a52,
        a53, a54, a63, a64, a65, so1, so2, baci, equs},
b = {0, b2, b3, b4, b5, 1/9};
c = {c1, c2, c2/(1 - 8*c2 + 20*c2^2),
     (16 - 25*c5 + 10*c2*(-5 + 8*c5))/(5*(5 - 8*c5 + 4*c2*(-4 + 7*c5))), c5, 1};
a = {{0, 0, 0, 0, 0, 0},
     {c2, 0, 0, 0, 0, 0},
     {c[[3]] - a32, a32, 0, 0, 0, 0},
     {c[[4]] - a42 - a43, a42, a43, 0, 0, 0},
     {c5 - a52 - a53 - a54, a52, a53, a54, 0, 0},
     {0, 1 - a63 - a64 - a65, a63, a64, a65, 0}};
e = {1, 1, 1, 1, 1, 1};
so1 = Solve[{b.c^3 == 1/4, b.c^2 == 1/3, b.c == 1/2, b.e == 1},
            {b2, b3, b4, b5}][[1]];
baci = Simplify[b.(a + DiagonalMatrix[c] - IdentityMatrix[6]) /. so1];
equs = Simplify[Join[baci[[2 ;; 5]],
                     {-(1/24) + b.a.a.c, -(1/24) + b.a.a.a.e},
                     {-(1/120) + b.a.a.a.c, -(1/120) + b.a.a.a.a.e, -(1/60) + b.a.a.c^2}] /. so1];
so2 = Solve[equs == {0, 0, 0, 0, 0, 0, 0, 0, 0},
                    {a32, a42, a43, a52, a53, a54, a63, a64, a65}][[1]];
Return[{a, b, c} /. Join[so1, so2]]]