<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(F_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be the free metabelian Lie algebra of rank <i>n</i> over a field <i>K</i> of characteristic zero. In this work, we provide an explicit set of generators and a presentation of generators of the algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((F_n')^{S_n}=F_n^{S_n}\cap F_n'\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mi>n</mi> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mi>n</mi> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> <mo>∩</mo> <msubsup> <mi>F</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> of symmetric polynomials as a right <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(K[X_n]^{S_n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <msup> <mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msup> </mrow> </math></EquationSource> </InlineEquation>-module. Besides, we compute the Hilbert series of the <i>K</i>-space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((F_n')^{S_n}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>F</mi> <mi>n</mi> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>