Natural extension of a Lie algebra A4,18b,|b|≤1, realizations, invariant systems and integrability

In the frame of Lie theory, the classical successive reduction of order for scalar ODEs is not much effective for system of differential equations due to technical hurdles. Lie algebraic approach is utilized to overcome these hurdles, then further invoked for the classification, linearization and integrability of system of differential equations. But in the case of higher dimension, this approach has also its own limitations due to classification of low dimensional Lie algebras and corresponding realizations. In the realm of Lie algebra theory, a natural extension refers to the process that involves enlarging a Lie algebra by adding new elements or structures while maintaining the fundamental algebraic properties. This extension technique finds applications in various fields, including physics, geometry, and mathematics.

In this paper, retaining $A_{4,18}^{b,N},\left|b\right|\le 1$ as a base algebra in $(1+2)$-dimensional space, examined in [2], a possible natural extension of underlying Lie algebra is investigated and also constructed associated realizations. Lie algebraic approach with natural extension is utilized for the analysis of a system of n second-order ODEs possessing extended form of Lie algebra $A_{4,18}^{b,N},\left|b\right|\le 1$ in $(1+2)$-dimensional space to $(1+n)$-dimensional space. In this research work, the extension of underlying Lie algebra, corresponding realizations and canonical forms associated with systems under consideration, have been formulated. Moreover, the integrability of investigated canonical forms for system of n second-order ODEs affiliated with extended form of Lie algebra $A_{4,18}^{b,N},\left|b\right|\le 1$ in $(1+2)$-dimensional space to $(1+n)$-dimensional space is described and three integrable classes of underlying systems have been deduced. In addition, two new classes of algebraic linearization criteria for a system of n second-order ODEs admitting Lie algebra $A_{2n,18}^{b,1},(b=0,-1)$ have been founded.

Novelty: To best of our knowledge, the natural extension of Lie algebra $A_{4,18}^{b,N},\left|b\right|\le 1$ and corresponding realizations in $(1+2)$-dimensional space to $(1+n)$-dimensional space are new contributions. Especially for underlying Lie algebra, three completely integrable classes and two new categorizations in linearization criteria for a system of n second-order ODEs are new contributions in literature and never been reported before this.