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We define the ring and the ideal:

sage: R.<x1,x2,x3> = GF(2)[]
sage: R
Multivariate Polynomial Ring in x1, x2, x3 over Finite Field of size 2
sage: I = R.ideal(x1*x2*x3-1, x2-x1, x1-1)

We compute the groebner basis of the ideal and we realize that these are exactly the right hand side of the three given equations:

sage: I.groebner_basis()
[x1 + 1, x2 + 1, x3 + 1]
sage: [x1-1, x2-1, x3-1]
[x1 + 1, x2 + 1, x3 + 1]

Thus, finding a solution to the 3 equations is equivalent to finding a solution to the following:

x1w2+x2w3I,x2w1+x3w3I,x1w1+x3w2I

which holds by definition of ideal for all elements w1,w2,w3 in the ideal I. Now, this does not include all solutions, for example:

sage: x1*x2 + x2*x3 in I
True
sage: x2 in I
False
sage: x3 in I
False

I stop here. Maybe you can continue finding all solutions using the groebner basis of the ideal computed above.

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No.2 Revision

We define the ring and the ideal:

sage: R.<x1,x2,x3> = GF(2)[]
sage: R
Multivariate Polynomial Ring in x1, x2, x3 over Finite Field of size 2
sage: I = R.ideal(x1*x2*x3-1, x2-x1, x1-1)

We compute the groebner basis of the ideal and we realize that these are exactly the right hand side of the three given equations:

sage: I.groebner_basis()
[x1 + 1, x2 + 1, x3 + 1]
sage: [x1-1, x2-1, x3-1]
[x1 + 1, x2 + 1, x3 + 1]

Thus, finding a solution to the 3 equations is equivalent to finding a solution to the following:

x1w2+x2w3I,x2w1+x3w3I,x1w1+x3w2I

which holds by definition of ideal for all elements w1,w2,w3 in the ideal I. I which solves the question as it was requested that the wi are in the ideal. Now, notice that this does not include all solutions, for example:

sage: x1*x2 + x2*x3 w2 = x2; w3 = x3
sage: w2 in I, w3 in I
(False, False)
sage: x1*w2 + x2*w3 in I
True
sage: x2 in I
False
sage: x3 in I
False

I stop here. Maybe you can continue finding all solutions using the groebner basis of the ideal computed above.