In this set of papers we formulate a stand alone method to derive maximal
number of linearizing transformations for nonlinear ordinary differential
equations (ODEs) of any order including coupled ones from a knowledge of fewer
number of integrals of motion. The proposed algorithm is simple,
straightforward and efficient and helps to unearth several new types of
linearizing transformations besides the known ones in the literature. To make
our studies systematic we divide our analysis into two parts. In the first part
we confine our investigations to the scalar ODEs and in the second part we
focuss our attention on a system of two coupled second order ODEs. In the case
of scalar ODEs, we consider second and third order nonlinear ODEs in detail and
discuss the method of deriving maximal number of linearizing transformations
irrespective of whether it is local or nonlocal type and illustrate the
underlying theory with suitable examples. As a by-product of this investigation
we unearth a new type of linearizing transformation in third order nonlinear
ODEs. Finally the study is extended to the case of general scalar ODEs. We then
move on to the study of two coupled second order nonlinear ODEs in the next
part and show that the algorithm brings out a wide variety of linearization
transformations. The extraction of maximal number of linearizing
transformations in every case is illustrated with suitable examples.Comment: Accepted for Publication in J. Nonlinear Math. Phys. (2012
