In this article, we concentrate on identifying all the non-zero, infinitely many integral solutions to the ternary cubic equation 2(l^2+m^2 )-3lm=56t^3. Of these solutions, some exciting patterns are discussed.
The universal language of the world is mathematics, which imparts knowledge of numbers, structures, formulas and shapes. Integers and integral valued functions are studied in the branch of pure mathematics known as Number theory. A polynomial equation with at least two unknowns that has only integer solutions is known as a Diophantine equation. The term ”Diophantine” refers to Diophantus of Alexandria, a third-century Hellenistic mathematician who studied these equations and was one of the first to introduce symbolism to algebra. Number theory is discussed in [3, 4, 9, 11] whereas in  Quadratic Diophantine equation is analysed. In [1, 2, 5, 7, 8, 10], the authors have considered cubic equation for study. In this work, a non homogeneous ternary cubic equation with three unknowns 2l2+m2-3lm=56t3 is considered in order to find some of its interesting integral solutions.
In this article, we have made an effort to obtain the integral solution of the non-homogeneous ternary cubic equation. Furthermore, one may search for another pattern of integral solution for the considered equation.
 Sharadha Kumar A. Vijayasankar and M.A. Gopalan. On non-homogeneous ternary cubic equation x3 + y3 + x + y = 2z(z2 ? ?2 ? 1). International Journal of Research Publication and Reviews, 2(8):592–598, 2021.
 John C. Butcher. A new solution to a cubic diophantine equation. Axioms, 11(5):184, 2022.
 R. D. Carmichael. The Theory of Numbers and Diophantine Analysis. Dover Publications Co., New York, 1950.
 L. E. Disckson. History of The Theory of Numbers, Volume II. Chelsia Publishing Co., New York, 1952.
 M.A Gopalan and Janaki .G. Integral solutions of (x2?y2)(3x2+3y2?2xy = 2(z2?w2)p3. Impact J Sci.,Tech, 4(1): 97–102, 2010.
 G. Janaki and Saranya. C. Observations on the ternary quadratic equation 6(x2 + y2) ? 11xy + 3x + 3y + 9 = 72z2. International Journal of Innovative Research in Science, Engineering and Technology, 5(2):2060–2065, Feb 2016.
 G. Janaki and Saranya. C. Integral solutions of the ternary cubic equation 3(x2 + y2) ? 4xy + 2(x + y + 1) = 972z3. International Research Journal of Engineering and Technology, 4(3):665–669, March 2017.
 G. Janaki and Saranya .P. On the ternary cubic diophantine equation 5(x2 +y2)? 6xy + 4(x + y) + 4 = 40z3. International Journal of Science and research-online, 5(3):227–229, March 2016.
 L.J. Mordell. Diophantine equations. Academic Press, London, 1969.
 K. Hema S. Vidhyalakshmi, J. Shanthi and M.A. Gopalan. Observation on the paper entitled intgeral solution of the homogeneous ternary cubic equation x3 + y3 = 52(x + y)z3. EPRA International Journal of Multidisciplinary Research (IJMR), 8(2), Feb 2022.
 S.G. Telang. Number Theory. Tata McGraw Hill publishing company,, New Delhi, 1996.