Exploring the symmetries of a complex mathematical equation, this paper presents an in-depth analysis of the (1+2)-dimensional fine Kolmogorov backward equation with a quadratic diffusivity. The study employs Lie invariance algebra and the direct method to compute the point symmetry pseudogroup of this equation. By classifying all subalgebras and analyzing its structure, the authors identify its essential subgroup and classify its discrete elements. This rigorous approach allows for the classification of Lie reductions and Lie invariant solutions, leading to the construction of wide families of solutions. These solutions are parameterized by arbitrary solutions of the (1+1)-dimensional linear heat equation or inverse square potentials. The research offers valuable insights into the symmetries and solutions of this intricate equation, advancing the field of applied mathematics and mathematical physics.
Published in Studies in Applied Mathematics, this research aligns with the journal's focus on original work in mathematics motivated by the application of mathematics to science and technology. By providing a detailed symmetry analysis of a complex ultraparabolic linear equation, the paper significantly contributes to the journal's scope of applied mathematics.