First, this paper provides characterizing the existence and uniqueness of the linear operator solution T for large classes of full Markov moment problems on closed subsets F of Rn. One uses approximation by special nonnegative polynomials. The case when F is compact is studied. Then the cases when F=Rn and F=R+n are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of Lν1(Rn), and respectively of Lν1(R+n), for a large class of measures ν. One solves the important difficulty created by the fact that on Rn, n≥2, there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples.