In this work, we search for infinitesimal equivalence transformations of 16 families of quasi-linear elliptic equations of the second order in two variables. A symmetry reduction has been performed for the families of equations u_tt + u_xx = f(t,x,u), u_tt + u_xx = f(x,u).

1. Zhalukevich D. S. Equivalence transformations of families of second-order quasi-linear hyperbolic equations of two variables. Arxiv [Preprint], 2025. Available at: https://www.researchgate.net/publication/398918025_Equivalence_transformations_of_families_of_second-order_quasi-linear_hyperbolic_equations_of_two_variables.

2. Zhalukevich D. S. Reduction of nonlinear Klein-Gordon equations for a function of two variables. Arxiv [Preprint], 2025. Available at: https://www.researchgate.net/publication/399149445_Reduction_of_nonlinear_Klein-Gordon_equations_for_a_function_of_two_variables.

3. Lahno V., Spichak S. V. Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition. Ukrainian Mathematical Journal, 2007, vol. 59, pp. 1719–1736. https://doi.org/10.1007/s11253-008-0021-z

4. Lahno V., Spichak S. V. Group classification of quasilinear elliptic-type equations. II. Invariance under solvable Lie algebras. Ukrainian Mathematical Journal, 2011, vol. 63, pp. 236–253. https://doi.org/10.1007/s11253-011-0501-4

5. Torrisi M., Tracina R., Valenti A. On equivalence transformations applied to a non-linear wave equation. Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics. Netherlands. Kluwer Academic Publ., 1993, pp. 367–375 https://doi.org/10.1007/978-94-011-2050-0_39

6. Tracina R. Invariants of a family of nonlinear wave equations. Communications in Nonlinear Science and Numerical Simulation. 2004, vol. 9, no. 1, pp. 127–133. https://doi.org/10.1016/S1007-5704(03)00021-2

7. Torrisi M., Tracina R., Valenti A. On the linearization of semilinear wave equations. Nonlinear Dynamics, 2004, vol. 36, pp. 97–106. https://doi.org/10.1023/b:nody.0000034649.74389.09

8. Boyko V. M., Lokaziuk O. V., Popovych R. O. Realizations of Lie algebras on the line and the new group classification of (1 + 1)-dimensional generalized nonlinear Klein – Gordon equations. Journal of Mathematical Physics, 2021, vol. 11, art. ID 127. https://doi.org/10.1007/s13324-021-00550-z

9. Lahno V., Magda O. The Group Classification of One Class of Nonlinear Wave Equations. Arxiv [Preprint], 2003. Available at: https://arxiv.org/abs/math-ph/0310049; https://doi.org/10.48550/arXiv.math-ph/0310049

10. Lahno V., Zhdanov R. Group classification of nonlinear wave equations. Journal of Mathematical Physics, 2005, vol. 46, no. 5, art. ID 053301. https://doi.org/10.1063/1.1884886

11. Lahno V., Zhdanov R., Magda O. Group classification and exact solutions of nonlinear wave equations. Acta Applicandae Mathematicae, 2006, vol. 91, pp. 253–313. https://doi.org/1010.1007/s10440-006-9039-0

12. Popovych R. O. Point and contact equivalence groupoids of two-dimensional quasilinear hyperbolic equations. Applied Mathematics Letters, 2021, vol. 116, art. ID 107068. https://doi.org/10.1016/j.aml.2021.107068

13. Lie S. Diskussion der Dierentialgleichung d2 z/dxdy = F(z). Archiv for Mathematik og Naturvidenskab, 1881, vol. 5, pp. 282–306.

14. Pucci E., Salvatori M. C. Group properties of a class of semilinear hyperbolic equations. International Journal of Non-Linear Mechanics, 1986, vol. 21, no. 2, pp. 147–155. https://doi.org/10.1016/0020-7462(86)90027-2

15. Azad H., Mustafa M. T., Ziad M. Group classification, optimal system and optimal reductions of a class of Klein Gordon equations. Communications in Nonlinear Science and Numerical Simulation, 2010, vol. 15, no. 5, pp. 1132–1147. https://doi.org/10.1016/j.cnsns.2009.05.045