  Let $L_n$ be the free metabelian Leibniz algebra generated by the set
$X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. This is the
free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz
algebras. We call an element $s(X_n)\in L_n$ symmetric if
$s(x_{\sigma(1)},\ldots,x_{\sigma(n)})=s(x_1,\ldots,x_n)$ for each permutation
$\sigma$ of $\{1,\ldots,n\}$. The set $L_n^{S_n}$ of symmetric polynomials of
$L_n$ is the algebra of invariants of the symmetric group $S_n$. Let $K[X_n]$
be the usual polynomial algebra with indeterminates from $X_n$. The description
of the algebra $K[X_n]^{S_n}$ is well known, and the algebra $(L_n')^{S_n}$ in
the commutator ideal $L_n'$ is a right $K[X_n]^{S_n}$-module. We give explicit
forms of elements of the $K[X_n]^{S_n}$-module $(L_n')^{S_n}$. Additionally, we
determine the description of the group ${\rm Inn}(L_{n}^{S_n})$ of inner
automorphisms of the algebra $L_n^{S_n}$. The findings can be considered as a
generalization of the recent results obtained for the free metabelian Lie
algebra of rank $n$.
