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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:

$$x_1w_2+x_2w_3 \in I, \quad x_2w_1+x_3w_3 \in I, \quad x_1w_1+x_3w_2 \in I$$

which holds by definition of ideal for all elements $w_1,w_2,w_3$ 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.

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:

$$x_1w_2+x_2w_3 \in I, \quad x_2w_1+x_3w_3 \in I, \quad x_1w_1+x_3w_2 \in I$$

which holds by definition of ideal for all elements $w_1,w_2,w_3$ in the ideal $I$. $I$ which solves the question as it was requested that the $w_i$ 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