<p>In this paper, we consider the <i>V</i>-soliton metric on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and on the total space of direct sum of fixed hermitian line bundle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^{\oplus n} \rightarrow M\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mo>⊕</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> and its projective compactification <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {P}(\mathbb {C} \oplus L^{\oplus n}) = \mathbb {P}(L^{-1} \oplus \mathbb {C}^{\oplus n})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">P</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo>⊕</mo> <msup> <mi>L</mi> <mrow> <mo>⊕</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="double-struck">P</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>⊕</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mo>⊕</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We prove that the <i>V</i>-soliton equation can be reduced to an ODE. The <i>V</i>-soliton metric is a solution to a degenerate fully nonlinear equation first introduced by La Nave and Tian in their study of Kähler-Ricci flow on symplectic quotients. This equation provides a framework for examining finite-time singularities of the Kähler-Ricci flow.</p>