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Yes, but not using solve.
Your system of equations defines a 0-dimensional subvariety of $\mathbb{C}^3$. Sage can tell you which points are in the zero set of the polynomials in your system like this:
sage: R.<x1,x2,x3> = PolynomialRing(QQ)
sage: R
Multivariate Polynomial Ring in x1, x2, x3 over Rational Field
sage: I=R.ideal([x1+x2+x3-6,x1*x2*x3-6,x1^2+x2^2+x3^2-14])
sage: I.dimension()
0
sage: I.variety()
[{x2: 2, x1: 3, x3: 1}, {x2: 3, x1: 2, x3: 1}, {x2: 1, x1: 3, x3: 2}, {x2: 3, x1: 1, x3: 2}, {x2: 1, x1: 2, x3: 3}, {x2: 2, x1: 1, x3: 3}]
The last line is a list of dictionaries. Each dictionary contains the coordinates of a point in the zero set of your system of polynomials.
