Recently the authors related to any symmetric function in two variables \[ f(x,y)=\sum m(\lambda_1,\lambda_2)S_{(\lambda_1,\lambda_2)}(x,y) \in {\mathbb C}[[x,y]], \] where \(S_{(\lambda_1,\lambda_2)}(x,y)\) is the Schur function associated with the partition \((\lambda_1,\lambda_2)\), its multiplicity series \(M(f)=\sum m(\lambda_1,\lambda_2) t^{\lambda_1}u^{\lambda_2}\); see V. Drensky and G. K. Genov [C. R. Acad. Bulg. Sci. 55, No. 6, 5–10 (2002; Zbl 1008.16023); J. Algebra 264, No. 2, 496–519 (2003; Zbl 1027.16013)]. In the present paper the authors announce the complete description of the symmetric functions \(f(x,y)\) with the property that \(M(f)\) is a rational function. This happens if and only if \(f\) is of the form \(f(x,y)=(p(x)+p(y))/q(x)q(y)\) for some polynomials \(p(x)\) and \(q(x)\) with coefficients from \({\mathbb C}(xy)\). The authors give also an effective procedure which calculates \(M(f)\). The main results are applied to invariant theory of unipotent linear transformations (or, equivalently, to linear locally nilpotent, or Weitzenböck, derivations) of polynomial algebras, presenting a simple method for calculation of the Hilbert series of the algebra of invariants, to the theory of PI-algebras, and to invariant theory of matrices.