<p>We prove that the period function of the center at the origin of the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2879_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}_k\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-equivariant differential equation <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2879_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="194" /> </InlineMediaObject> <EquationSource Format="TEX">\(\dot{z}=iz+a(z\overline{z})^nz^{k+1}, a\ne 0,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>z</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>i</mi> <mi>z</mi> <mo>+</mo> <mi>a</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mover> <mi>z</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <msup> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> is monotonous decreasing for all <i>n</i> and <i>k</i> positive integers, solving a conjecture about them. We show this result as corollary of proving that the period function of the center at the origin of a sub-family of the reversible quadratic centers is monotonous decreasing as well.</p>