Estimates of the fundamental solution to the Cauchy problem have been obtained for a second-order ultraparabolic equation with two groups of spatial variables of degeneracy and growing coefficients depending on the time variable and the parametric variable of the main group of spatial variables. Formulas for the Lie derivative of the volume potential generated by this fundamental solution of the Cauchy problem have been established. The resulting expressions can be utilized in constructing the fundamental solution to the Cauchy problem for a second-order ultraparabolic equation with two groups of variables of degeneracy and growing coefficients.

[1] Di Francesco, M., Pascucci, A. On a class of degenerate parabolic equations of Kolmogorov type Applied Mathematics Research eXpress. 2005. (3), 77-116. https://doi.org/10.1155/AMRX.2005.77

[2] Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Birkh¨auser, Basel, 2004. (Ser. Operator Theory: Adv. and Appl. 152). htts://doi.org/10.1007/978-3-0348-7844-9

[3] Ivasyshen S. D., Layuk V. V. Fundamental solutions of the Cauchy problem for some degenerate parabolic equations of Kolmogorov type. Ukr. Mat. J. 2011, 63 (11), 1469–1500. (in Ukrainian)

[4] Ivasyshen S. D., Medyns’kyi I.P. On the classical fundamental solutions of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables. J. Math. Sci. 2018. 231 (4), 507–526. https:doi.org//10.1007/s10958-018-3830-0.

[5] Ivasyshen S. D., Medynsky I.P. Classical fundamental solutions of the Cauchy problem for ultraparabolic Kolmogorov-type equations with two groups of spatial variables of degeneration. II. J. Math. Sci. 2020. 247 (1), 1–23. https:doi.org//10.1007/s10958-020-04786-1.

[6] Medinskyi I., Pasichnyk H. Fundamental solution to Cauchy problem for kolmogorov-type equation with unbounded coefficients that do not depend on degeneration variables. Reports of the National Academy of Sciences of Ukraine. 2025, (3), 3–16.  https://doi.org/10.15407/dopovidi2025.03.003 (in Ukrainian)

[7] Pasichnyk H. Fundamental solution of the Cauсhy problem for the dissipative ultraparabolic Kolmogorov equation with coefficients independent on variable degeneration. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics. 2025, 81(2), 130-137. https://doi.org/10.17721/1812-5409.2025/2.20 (in Ukrainian)