  The linearization problem for nonlinear second-order ODEs to the Laguerre
form by means of generalized Sundman transformations (S-transformations) is
considered, which has been investigated by Duarte et al. earlier. A
characterization of these S-linearizable equations in terms of first integral
and procedure for construction of linearizing S-transformations has been given
recently by Muriel and Romero. Here we give a new characterization of
S-linearizable equations in terms of the coefficients of ODE and one auxiliary
function. This new criterion is used to obtain the general solutions for the
first integral explicitly, providing a direct alternative procedure for
constructing the first integrals and Sundman transformations. The effectiveness
of this approach is demonstrated by applying it to find the general solution
for geodesics on surfaces of revolution of constant curvature in a unified
manner.
