Five equivalence classes had been found for systems of two second-order
ordinary differential equations, transformable to linear equations
(linearizable systems) by a change of variables. An "optimal (or simplest)
canonical form" of linear systems had been established to obtain the symmetry
structure, namely with 5, 6, 7, 8 and 15 dimensional Lie algebras. For those
systems that arise from a scalar complex second-order ordinary differential
equation, treated as a pair of real ordinary differential equations, a "reduced
optimal canonical form" is obtained. This form yields three of the five
equivalence classes of linearizable systems of two dimensions. We show that
there exist 6, 7 and 15-dimensional algebras for these systems and illustrate
our results with examples
