Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 214872, 9 pages http://dx.doi.org/10.1155/2013/214872 Research Article Characterization of Symmetry Properties of First Integrals for Submaximal Linearizable Third-Order ODEs K. S. Mahomed and E. Momoniat Diferential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa Correspondence should be addressed to K. S. Mahomed; komalmajeed@hotmail.com Received 23 August 2013; Accepted 13 September 2013 Academic Editor: Chaudry Masood Khalique Copyright © 2013 K. S. Mahomed and E. Momoniat. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te relationship between frst integrals of submaximal linearizable third-order ordinary diferential equations (ODEs) and their symmetries is investigated. We obtain the classifying relations between the symmetries and the frst integral for submaximal cases of linear third-order ODEs. It is known that the maximum Lie algebra of the frst integral is achieved for the simplest equation and is four-dimensional. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the frst integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the frst integrals can have 0, 1, 2, and 3 symmetries, and for the 4 symmetry class of linear third-order ODEs, they are 0, 1, and 2 symmetries, respectively. In the case of submaximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals. 1. Introduction Algebraic properties of frst integrals of scalar-order difer- ential equations have been of interest in the recent literature since the early works of Lie [1, 2] on symmetries and invariants of ODEs. Te Noether classifcation has also drawn attention to them in [3]. Te symmetry classifcation of scalar ordinary diferential equations has been studied in recent years (see, e.g., [4, 5]). Of the algebraic properties, the max- imal symmetry properties of frst integrals of linear ODEs have attracted particular attention. In [6], the authors showed that the full Lie algebra sl(3,) of scalar linear second-order ODEs represented by the simplest free particle equation can be generated by the three triplets of the three-dimensional algebras of the two basic integrals and their quotient. In their work [4], they found that the symmetries of the maximal cases of scalar linear th-order ODEs, ≥3, are +1, +2, and +4. Tus, for scalar linear third-order equations these correspond to 4, 5, and 7 symmetries. Govinder and Leach studied the symmetry properties of frst integrals of scalar linear third-order ODEs which belong to these three classes in [7]. Tey showed that the three equivalence classes each has certain frst integrals with a specifc number of point symme- tries. Later Flessas et al. in [8] examined the symmetry struc- ture of the frst integrals of higher-order equations of maxi- mal symmetry and they proved some interesting basic propo- sitions related to the scaling symmetry and basic integrals. In a recent paper [9], Mahomed and Momoniat, obtained a classifying relation between the symmetries and the frst integrals of linear or linearizable scalar second-order ODEs. Tey presented a complete classifcation of point symmetries of frst integrals of such linear ODEs, and as a consequence, they provided a counting theorem for the point symmetries of frst integrals of scalar linearizable second-order ODEs. Tey showed that there exist the 0, 1, 2, or 3 point symmetry cases and that the maximal algebra case is unique. Tese authors then considered the problem of classifying the symmetry property of the frst integrals of the simplest third- order equation   =0 in the paper [10]. Tey found that the maximal Lie algebra of a frst integral for this equation is unique and four-dimensional. Tey also showed that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of two basic integrals instead of three. Moreover, they obtained counting theorems of