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Try this:

R = PolynomialRing(GF(2),3,"xyz")
x,y,z = R.gens()
S.<a,b,c> = R.quotient((x^2 + 1,y^2+1,z^2+1))
K = S[var("u v")]
q = K.gens()

Pol = (q[1]*K(a)+q[0])*(K(a)*K(b))-5*K(b)*K(a) # Equation to solve

I = S.ideal(Pol.coefficients())
I.variety()