It is a singular quadric with a center [1,1,0] in original coordinates, as Panchatcharam wrote. This is also the unique real solution.
Theoretical framework:
The corresponding matrices A4x4 and B3x3 (enclosed) have determinants det(A) = 0 and det(B) = 1, all eigenvalues of B are positive.
Thus the equation can be transformed in X2/a2+Y2/b2+Z2/c2 = 0.
The transformed equation (numerical values) is enclosed as well.
Dear Leonid,
maybe is this an equation of a tumor. The equation is that of an ellipsoid [reduced in 1 point, its center]. A small change in one parameter, say the linear term -2x replaced by -(2+d)x, would give a non-reduced ellipsoid.
Yes. It is confusing me. Since he mentioned "finite real solutions", I thought that he is expecting set of all (x,y,z) in R3 which satisfies the equation. So, it may be better to rephrase his question to avoid the ambiguity of the word "solve".