<p>The purpose of this paper is to develop the theory of holomorphic functions with modulus bounded by 1 on the symmetrized skew bidisc <Equation ID="Equ96"> <EquationSource Format="TEX">\( \mathbb {G}_{r} {\mathop {=}\limits ^\textrm{def}} \Big \{( \lambda _{1}+r\lambda _{2} ,r\lambda _{1}\lambda _{2}): \lambda _{1}\in \mathbb {D}, \lambda _{2}\in \mathbb {D}\Big \}, \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="double-struck">G</mi> <mi>r</mi> </msub> <mover> <mo>=</mo> <mtext>def</mtext> </mover> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>r</mi> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>r</mi> <msub> <mi>λ</mi> <mn>1</mn> </msub> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">D</mi> <mo>,</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">D</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>for a fixed <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(r \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show the existence of a realization formula and a model formula for such holomorphic functions.</p>