We provide a supplementation of the results on the canonical forms for scalar fourthorder ordinary differential equations (ODEs) which admit fourdimensional Lie algebras obtained recently. Together with these new canonical forms, a complete list of scalar fourthorder ODEs that admit fourdimensional Lie algebras is available.
The integrability of scalar ordinary differential equations (ODEs) by use of the Lie symmetry method depends on their symmetrical Lie algebra if the Lie algebra is solvable and of sufficient dimension. There exists two different approaches to the integrability of differential equations using Lie point symmetries. One is the direct method in which Lie point symmetries are utilized to perform integrability by successive reduction of order of the equation using ideals of the algebra. The other approach is the canonical form method if the equations are classified into different types according to the canonical forms of the corresponding Lie algebra.
Lie [
The canonical forms for scalar thirdorder ODEs that admit three symmetries were obtained by Mahomed and Leach [
In their paper, Cerquetelli et al. [
Apart from scalar fourthorder ODEs arising in the symmetry reductions of partial differential equations such as the linear wave equation in an inhomogeneous medium (see [
Firstly, we provide a comparison of the results of [
We show here that the results on realizations of fourdimensional algebras in the plane given in [
missing of some inequivalent cases and
mutually equivalent cases.
In the following comparison, some first type of errors exist in [
We use the nomenclature of Patera and Winternitz [
Here the
Lie algebra 

Realizations (generators) 


11 


5 

*7 



9 


9 


8 


*8 

9 



8 


8 


8 


7 


7 

10 



6 


5 


5 

*7 



*6 

7 

Lie algebra 

Realizations and equations 


*7 




*8 




*7 




*6 


In this contribution we have supplemented the work [
A. Fatima gratefully acknowledges the financial support and scholarship from School of Computational and Applied Mathematics, University of the Witwatersrand.