<p>We will address the questions and conjectures left by the recent papers by Ferreira et al. (Bollettino dell’Unione Matematica Italiana, 2023, <a href="https://doi.org/10.1007/s40574-023-00402-7">https://doi.org/10.1007/s40574-023-00402-7</a>), (J Algebra 638:488–505, 2024). We also present an example showing the necessity of the conditions of the results that answer the conjectures. It leads us to the generalized Vukman equation: <Equation ID="Equ36"> <EquationSource Format="TEX">\(\begin{aligned} f(x)+x^ng(x^{-1})=0, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>x</mi> <mi>n</mi> </msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for every invertible <i>x</i>, where <i>n</i> is a nonnegative integer and <i>f</i>, <i>g</i> are additive maps on an alternative ring <i>D</i>. We will study this equation for the split octonion algebras, the alternative division rings, and the field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {Z}}_2(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of rational functions over the field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {Z}}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> with two elements.</p>