# Great overdetermined system of quadratic equations.

Could anyone help solve the system of 81 quadratic equations with 27 variables? The system has solutions. Restriction:
$$sum_ {i = 0} ^ {26} x ^ 2_i = 1$$

$$-x_1 x_3-x_2 x_6 + x_1 x_9 + x_2 x_ {18} = 0$$

$$-x_1 x_4-x_2 x_7 + x_1 x_ {10} + x_2 x_ {19} = 0$$

$$-x_1 x_5-x_2 x_8 + x_1 x_ {11} + x_2 x_ {20} = 0$$

$$-x_3 x_4-x_5 x_6 + x_1 x_ {12} + x_2 x_ {21} = 0$$

$$-x_1 x_3 + x_0 x_4-x_4 ^ 2-x_5 x_7 + x_1 x_ {13} + x_2 x_ {22} = 0$$

$$-x_2 x_3 + x_0 x_5-x_4 x_5-x_5 x_8 + x_1 x_ {14} + x_2 x_ {23} = 0$$

$$-x_3 x_7-x_6 x_8 + x_1 x_ {15} + x_2 x_ {24} = 0$$

$$-x_1 x_6 + x_0 x_7-x_4 x_7-x_7 x_8 + x_1 x_ {16} + x_2 x_ {25} = 0$$

$$-x_2 x_6-x_5 x_7 + x_0 x_8-x_8 ^ 2 + x_1 x_ {17} + x_2 x_ {26} = 0$$

$$x_0 x_3-x_0 x_9 + x_4 x_9-x_3 x_ {10} -x_6 x_ {11} + x_5 x_ {18} = 0$$

$$x_1 x_3-x_1 x_9-x_7 x_ {11} + x_5 x_ {19} = 0$$

$$x_2 x_3-x_2 x_9-x_5 x_ {10} + x_4 x_ {11} -x_8 x_ {11} + x_5 x_ {20} = 0$$

$$x_3 ^ 2-x_0 x_ {12} + x_4 x_ {12} -x_3 x_ {13} -x_6 x_ {14} + x_5 x_ {21} = 0$$

$$x_3 x_4-x_1 x_ {12} -x_7 x_ {14} + x_5 x_ {22} = 0$$

$$x_3 x_5-x_2 x_ {12} -x_5 x_ {13} + x_4 x_ {14} -x_8 x_ {14} + x_5 x_ {23} = 0$$

$$x_3 x_6-x_0 x_ {15} + x_4 x_ {15} -x_3 x_ {16} -x_6 x_ {17} + x_5 x_ {24} = 0$$

$$x_3 x_7-x_1 x_ {15} -x_7 x_ {17} + x_5 x_ {25} = 0$$

$$x_3 x_8-x_2 x_ {15} -x_5 x_ {16} + x_4 x_ {17} -x_8 x_ {17} + x_5 x_ {26} = 0$$

$$x_0 x_6 + x_7 x_9-x_0 x_ {18} + x_8 x_ {18} -x_3 x_ {19} -x_6 x_ {20} = 0$$

$$x_1 x_6 + x_7 x_ {10} -x_1 x_ {18} -x_4 x_ {19} + x_8 x_ {19} -x_7 x_ {20} = 0$$

$$x_2 x_6 + x_7 x_ {11} -x_2 x_ {18} -x_5 x_ {19} = 0$$

$$x_3 x_6 + x_7 x_ {12} -x_0 x_ {21} + x_8 x_ {21} -x_3 x_ {22} -x_6 x_ {23} = 0$$

$$x_4 x_6 + x_7 x_ {13} -x_1 x_ {21} -x_4 x_ {22} + x_8 x_ {22} -x_7 x_ {23} = 0$$

$$x_5 x_6 + x_7 x_ {14} -x_2 x_ {21} -x_5 x_ {22} = 0$$

$$x_6 ^ 2 + x_7 x_ {15} -x_0 x_ {24} + x_8 x_ {24} -x_3 x_ {25} -x_6 x_ {26} = 0$$

$$x_6 x_7 + x_7 x_ {16} -x_1 x_ {24} -x_4 x_ {25} + x_8 x_ {25} -x_7 x_ {26} = 0$$

$$x_6 x_8 + x_7 x_ {17} -x_2 x_ {24} -x_5 x_ {25} = 0$$

$$x_9 x_ {10} -x_1 x_ {12} -x_2 x_ {15} + x_ {11} x_ {18} = 0$$

$$x_1 x_9-x_0 x_ {10} + x_ {10} ^ 2-x_1 x_ {13} -x_2 x_ {16} + x_ {11} x_ {19} = 0$$

$$x_2 x_9-x_0 x_ {11} + x_ {10} x_ {11} -x_1 x_ {14} -x_2 x_ {17} + x_ {11} x_ {20} = 0$$

$$-x_4 x_ {12} + x_ {10} x_ {12} -x_5 x_ {15} + x_ {11} x_ {21} = 0$$

$$x_4 x_9-x_3 x_ {10} -x_4 x_ {13} + x_ {10} x_ {13} -x_5 x_ {16} + x_ {11} x_ {22} = 0$$

$$x_5 x_9-x_3 x_ {11} -x_4 x_ {14} + x_ {10} x_ {14} -x_5 x_ {17} + x_ {11} x_ {23} = 0$$

$$-x_7 x_ {12} -x_8 x_ {15} + x_ {10} x_ {15} + x_ {11} x_ {24} = 0$$

$$x_7 x_9-x_6 x_ {10} -x_7 x_ {13} -x_8 x_ {16} + x_ {10} x_ {16} + x_ {11} x_ {25} = 0$$

$$x_8 x_9-x_6 x_ {11} -x_7 x_ {14} -x_8 x_ {17} + x_ {10} x_ {17} + x_ {11} x_ {26} = 0$$

$$-x_9 ^ 2 + x_0 x_ {12} -x_ {10} x_ {12} + x_9 x_ {13} -x_ {11} x_ {15} + x_ {14} x_ {18} = 0$$

$$-x_9 x_ {10} + x_1 x_ {12} -x_ {11} x_ {16} + x_ {14} x_ {19} = 0$$

$$-x_9 x_ {11} + x_2 x_ {12} + x_ {11} x_ {13} -x_ {10} x_ {14} -x_ {11} x_ {17} + x_ {14} x_ {20} = 0$$

$$x_3 x_ {12} -x_9 x_ {12} -x_ {14} x_ {15} + x_ {14} x_ {21} = 0$$

$$x_4 x_ {12} -x_ {10} x_ {12} -x_ {14} x_ {16} + x_ {14} x_ {22} = 0$$

$$x_5 x_ {12} -x_ {11} x_ {12} -x_ {14} x_ {17} + x_ {14} x_ {23} = 0$$

$$x_6 x_ {12} -x_9 x_ {15} + x_ {13} x_ {15} -x_ {12} x_ {16} -x_ {15} x_ {17} + x_ {14} x_ {24} = 0$$

$$x_7 x_ {12} -x_ {10} x_ {15} -x_ {16} x_ {17} + x_ {14} x_ {25} = 0$$

$$x_8 x_ {12} -x_ {11} x_ {15} -x_ {14} x_ {16} + x_ {13} x_ {17} -x_ {17} ^ 2 + x_ {14} x_ {26} = 0$$

$$x_0 x_ {15} + x_9 x_ {16} -x_9 x_ {18} + x_ {17} x_ {18} -x_ {12} x_ {19} -x_ {15} x_ {20} = 0$$

$$x_1 x_ {15} + x_ {10} x_ {16} -x_ {10} x_ {18} -x_ {13} x_ {19} + x_ {17} x_ {19} -x_ {16} x_ {20 } = 0$$

$$x_2 x_ {15} + x_ {11} x_ {16} -x_ {11} x_ {18} -x_ {14} x_ {19} = 0$$

$$x_3 x_ {15} + x_ {12} x_ {16} -x_9 x_ {21} + x_ {17} x_ {21} -x_ {12} x_ {22} -x_ {15} x_ {23} = 0$$

$$x_4 x_ {15} + x_ {13} x_ {16} -x_ {10} x_ {21} -x_ {13} x_ {22} + x_ {17} x_ {22} -x_ {16} x_ {23 } = 0$$

$$x_5 x_ {15} + x_ {14} x_ {16} -x_ {11} x_ {21} -x_ {14} x_ {22} = 0$$

$$x_6 x_ {15} + x_ {15} x_ {16} -x_9 x_ {24} + x_ {17} x_ {24} -x_ {12} x_ {25} -x_ {15} x_ {26} = 0$$

$$x_7 x_ {15} + x_ {16} ^ 2-x_ {10} x_ {24} -x_ {13} x_ {25} + x_ {17} x_ {25} -x_ {16} x_ {26} = 0$$

$$x_8 x_ {15} + x_ {16} x_ {17} -x_ {11} x_ {24} -x_ {14} x_ {25} = 0$$

$$x_9 x_ {19} + x_ {18} x_ {20} -x_1 x_ {21} -x_2 x_ {24} = 0$$

$$x_1 x_ {18} -x_0 x_ {19} + x_ {10} x_ {19} + x_ {19} x_ {20} -x_1 x_ {22} -x_2 x_ {25} = 0$$

$$x_2 x_ {18} + x_ {11} x_ {19} -x_0 x_ {20} + x_ {20} ^ 2-x_1 x_ {23} -x_2 x_ {26} = 0$$

$$x_ {12} x_ {19} -x_4 x_ {21} + x_ {20} x_ {21} -x_5 x_ {24} = 0$$

$$x_4 x_ {18} -x_3 x_ {19} + x_ {13} x_ {19} -x_4 x_ {22} + x_ {20} x_ {22} -x_5 x_ {25} = 0$$

$$x_5 x_ {18} + x_ {14} x_ {19} -x_3 x_ {20} -x_4 x_ {23} + x_ {20} x_ {23} -x_5 x_ {26} = 0$$

$$x_ {15} x_ {19} -x_7 x_ {21} -x_8 x_ {24} + x_ {20} x_ {24} = 0$$

$$x_7 x_ {18} -x_6 x_ {19} + x_ {16} x_ {19} -x_7 x_ {22} -x_8 x_ {25} + x_ {20} x_ {25} = 0$$

$$x_8 x_ {18} + x_ {17} x_ {19} -x_6 x_ {20} -x_7 x_ {23} -x_8 x_ {26} + x_ {20} x_ {26} = 0$$

$$-x_9 x_ {18} + x_0 x_ {21} -x_ {10} x_ {21} + x_9 x_ {22} + x_ {18} x_ {23} -x_ {11} x_ {24} = 0$$

$$-x_9 x_ {19} + x_1 x_ {21} + x_ {19} x_ {23} -x_ {11} x_ {25} = 0$$

$$-x_9 x_ {20} + x_2 x_ {21} + x_ {11} x_ {22} -x_ {10} x_ {23} + x_ {20} x_ {23} -x_ {11} x_ {26} = 0$$

$$-x_ {12} x_ {18} + x_3 x_ {21} -x_ {13} x_ {21} + x_ {12} x_ {22} + x_ {21} x_ {23} -x_ {14} x_ { 24} = 0$$

$$-x_ {12} x_ {19} + x_4 x_ {21} + x_ {22} x_ {23} -x_ {14} x_ {25} = 0$$

$$-x_ {12} x_ {20} + x_5 x_ {21} + x_ {14} x_ {22} -x_ {13} x_ {23} + x_ {23} ^ 2-x_ {14} x_ {26} = 0$$

$$-x_ {15} x_ {18} + x_6 x_ {21} -x_ {16} x_ {21} + x_ {15} x_ {22} -x_ {17} x_ {24} + x_ {23} x_ { 24} = 0$$

$$-x_ {15} x_ {19} + x_7 x_ {21} -x_ {17} x_ {25} + x_ {23} x_ {25} = 0$$

$$-x_ {15} x_ {20} + x_8 x_ {21} + x_ {17} x_ {22} -x_ {16} x_ {23} -x_ {17} x_ {26} + x_ {23} x_ { 26} = 0$$

$$-x_ {18} ^ 2-x_ {19} x_ {21} + x_0 x_ {24} -x_ {20} x_ {24} + x_9 x_ {25} + x_ {18} x_ {26} = 0$$

$$-x_ {18} x_ {19} -x_ {19} x_ {22} + x_1 x_ {24} + x_ {10} x_ {25} -x_ {20} x_ {25} + x_ {19} x_ { 26} = 0$$

$$-x_ {18} x_ {20} -x_ {19} x_ {23} + x_2 x_ {24} + x_ {11} x_ {25} = 0$$

$$-x_ {18} x_ {21} -x_ {21} x_ {22} + x_3 x_ {24} -x_ {23} x_ {24} + x_ {12} x_ {25} + x_ {21} x_ { 26} = 0$$

$$-x_ {19} x_ {21} -x_ {22} ^ 2 + x_4 x_ {24} + x_ {13} x_ {25} -x_ {23} x_ {25} + x_ {22} x_ {26} = 0$$

$$-x_ {20} x_ {21} -x_ {22} x_ {23} + x_5 x_ {24} + x_ {14} x_ {25} = 0$$

$$x_6 x_ {24} -x_ {18} x_ {24} + x_ {15} x_ {25} -x_ {21} x_ {25} = 0$$

$$x_7 x_ {24} -x_ {19} x_ {24} + x_ {16} x_ {25} -x_ {22} x_ {25} = 0$$

$$x_8 x_ {24} -x_ {20} x_ {24} + x_ {17} x_ {25} -x_ {23} x_ {25} = 0$$

``````Do[{i, j, k} = {0, 0, IntegerDigits[l, 3][[1]]};
Co[i, j, k] = Subindex[x, l], {l, 0, 2, 1}]Do[{i, j, k} = {0, IntegerDigits[l, 3][[1]],
Whole digits[l, 3][[2]]};
Co[i, j, k] = Subindex[x, l]{1, 3, 8, 1}]Do[{ijk}={IntegerDigits[{ijk}={IntegerDigits[{ijk}={IntegerDigits[{ijk}={IntegerDigits[l, 3][[1]], IntegerDigits[l, 3][[2]],
Whole digits[l, 3][[3]]};
Co[i, j, k] = Subindex[x, l], {l, 9, 26, 1}]Do[lam = j 3^3 + k 3^2 + n 3 + s;
Subscript[eq, lam] =
Sum[Co[j, k, m]*Co[m, n, s] - Co[k, n, m]*Co[j, m, s]{m, 0, 2,
one}]{j, 0, 2, 1}, {k, 0, 2, 1}, {n, 0, 2, 1}, {s, 0, 2, 1}]Solve[
Subscript[eq, 1] == 0 && Subindex[eq, 2] == 0 &&
Subscript[eq, 3] == 0 && Subindex[eq, 4] == 0 &&
Subscript[eq, 5] == 0 && Subindex[eq, 6] == 0 &&
Subscript[eq, 7] == 0 && Subindex[eq, 8] == 0 &&
Subscript[eq, 9] == 0 && Subindex[eq, 10] == 0 &&
Subscript[eq, 11] == 0 && Subindex[eq, 12] == 0 &&
Subscript[eq, 13] == 0 && Subindex[eq, 14] == 0 &&
Subscript[eq, 15] == 0 && Subindex[eq, 16] == 0 &&
Subscript[eq, 17] == 0 && Subindex[eq, 18] == 0 &&
Subscript[eq, 19] == 0 && Subindex[eq, 20] == 0 &&
Subscript[eq, 21] == 0 && Subindex[eq, 22] == 0 &&
Subscript[eq, 23] == 0 && Subindex[eq, 24] == 0 &&
Subscript[eq, 25] == 0 && Subindex[eq, 26] == 0 &&
Subscript[eq, 27] == 0 && Subindex[eq, 28] == 0 &&
Subscript[eq, 29] == 0 && Subindex[eq, 30] == 0 &&
Subscript[eq, 31] == 0 && Subindex[eq, 32] == 0 &&
Subscript[eq, 33] == 0 && Subindex[eq, 34] == 0 &&
Subscript[eq, 35] == 0 && Subindex[eq, 36] == 0 &&
Subscript[eq, 37] == 0 && Subindex[eq, 38] == 0 &&
Subscript[eq, 39] == 0 && Subindex[eq, 40] == 0 &&
Subscript[eq, 41] == 0 && Subindex[eq, 42] == 0 &&
Subscript[eq, 43] == 0 && Subindex[eq, 44] == 0 &&
Subscript[eq, 45] == 0 && Subindex[eq, 46] == 0 &&
Subscript[eq, 47] == 0 && Subindex[eq, 48] == 0 &&
Subscript[eq, 49] == 0 && Subindex[eq, 50] == 0 &&
Subscript[eq, 51] == 0 && Subindex[eq, 52] == 0 &&
Subscript[eq, 53] == 0 && Subindex[eq, 54] == 0 &&
Subscript[eq, 55] == 0 && Subindex[eq, 56] == 0 &&
Subscript[eq, 57] == 0 && Subindex[eq, 58] == 0 &&
Subscript[eq, 59] == 0 && Subindex[eq, 60] == 0 &&
Subscript[eq, 61] == 0 && Subindex[eq, 62] == 0 &&
Subscript[eq, 63] == 0 && Subindex[eq, 64] == 0 &&
Subscript[eq, 65] == 0 && Subindex[eq, 66] == 0 &&
Subscript[eq, 67] == 0 && Subindex[eq, 68] == 0 &&
Subscript[eq, 69] == 0 && Subindex[eq, 70] == 0 &&
Subscript[eq, 71] == 0 && Subindex[eq, 72] == 0 &&
Subscript[eq, 73] == 0 && Subindex[eq, 74] == 0 &&
Subscript[eq, 75] == 0 && Subindex[eq, 76] == 0 &&
Subscript[eq, 77] == 0 && Subindex[eq, 78] == 0 &&
Subscript[eq, 79] == 0 && Subindex[eq, 80] == 0 &&
Subscript[eq, 0] == 0
, {Subindex[x, 0], Subscript[x, 1], Subscript[x, 2], Subscript[x,
3], Subscript[x, 4], Subscript[x, 5], Subscript[x, 6], Subscript[x,
7], Subscript[x, 8], Subscript[x, 9], Subscript[x, 10], Subscript[x,
11], Subscript[x, 12], Subscript[x, 13], Subscript[x, 14],
Subscript[x, 15], Subscript[x, 16], Subscript[x, 17], Subscript[x,
18], Subscript[x, 19], Subscript[x, 20], Subscript[x, 21],
Subscript[x, 22], Subscript[x, 23], Subscript[x, 24], Subscript[x,
25], Subscript[x, 26]}]
``````