The explicit solutions of previously given determining equations for isovector components (infinitesimal generators) associated with Lie groups of equivalence transformations for the most general family of second order balance equations involving finite arbitrary number of independent and dependent variables are obtained. Equivalence transformations which are considered in this work map the solutions of partial differential equations containing arbitrary functions or parameters to the solutions of the equations of the same structure but with different functions or parameters. The general solutions discovered here are employed to determine the isovector fields corresponding (i) to a one-imensional non-linear wave equation, (ii) to the field equations of non-linear homogeneous hyperelasticity. These isovector fields were found in earlier works by solving determining equations associated with the each individual case. (C) 2000 Elsevier Science Ltd. All rights reserved.