<p>We study piecewise-smooth systems with three zones, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\dot{z} = f_i(z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>z</mi> <mo>˙</mo> </mover> <mo>=</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(i = 1,2,3,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> whose discontinuity set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> consists either of a pair of parallel lines or a pair of circles tangent to each other internally or externally. Each <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(f_i:\overline{\mathbb {C}} \rightarrow \overline{\mathbb {C}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>:</mo> <mover> <mi mathvariant="double-struck">C</mi> <mo>¯</mo> </mover> <mo stretchy="false">→</mo> <mover> <mi mathvariant="double-struck">C</mi> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> is assumed to be a holomorphic function. We establish conditions ensuring the existence of limit cycles in such systems and provide lower bounds for the maximum number of limit cycle. Our approach combines the Melnikov method, local integrability properties of holomorphic systems, and the existence of normal forms around zeros and poles.</p>