The realization space is
  [1   1   0                    1   0   1   1                   0                    1         1    1]
  [0   1   1   -3*x1^2 + 4*x1 - 1   0   0   1                   1   -3*x1^2 + 4*x1 - 1        x1   x1]
  [0   0   0                    0   1   1   1   3*x1^2 - 5*x1 + 3                   x1   -x1 + 1   x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (3*x1^3 - 5*x1^2 + 4*x1 - 1, -27*x1^7 + 117*x1^6 - 219*x1^5 + 222*x1^4 - 130*x1^3 + 42*x1^2 - 6*x1)
avoiding the zero loci of the polynomials
RingElem[x1, 2*x1 - 1, x1 - 1, 3*x1^2 - 3*x1 + 1, 3*x1^2 - 5*x1 + 3, 3*x1 - 2, 3*x1^3 - 8*x1^2 + 7*x1 - 3, 3*x1^3 - 5*x1^2 + 2*x1 + 1, 3*x1 - 1, 3*x1^3 - 7*x1^2 + 4*x1 - 1, 3*x1^2 - 4*x1 + 2, 3*x1^3 - 8*x1^2 + 5*x1 - 1, 9*x1^4 - 24*x1^3 + 27*x1^2 - 12*x1 + 2, 3*x1^2 - 5*x1 + 4, 3*x1^3 - 6*x1^2 + 5*x1 - 1, 3*x1^3 - 8*x1^2 + 9*x1 - 3, 3*x1^3 - 4*x1^2 + 3*x1 - 1, 9*x1^4 - 27*x1^3 + 32*x1^2 - 16*x1 + 3, 9*x1^4 - 27*x1^3 + 35*x1^2 - 21*x1 + 6, 9*x1^3 - 18*x1^2 + 14*x1 - 2, 9*x1^4 - 27*x1^3 + 35*x1^2 - 21*x1 + 5, 9*x1^3 - 18*x1^2 + 17*x1 - 5, 9*x1^4 - 27*x1^3 + 32*x1^2 - 17*x1 + 2]