An effective computational scheme to construct the adiabatic potential energy surfaces (APES) along the reaction coordinates for an electron transfer reaction occurring by two steps at a metal electrode is considered in the framework of the Anderson-Newns model. Two Theorems have been proved which predict the existence of all possible solutions of the Anderson-Newns equations at arbitrary values of the key parameters and point out the region for each solution. Asymptotic formulas for solutions near multiple roots have been derived and combined in an effective way with numerical routines. The analysis of some important properties of the APES, which can be of interest for modelling the electrochemical electron transfer processes, is presented as well. The APES describing the reduction of Zn(II) and In(111) aqua-complexes at a mercury electrode have been built and discussed. (C) 2004 Elsevier B.V. All rights reserved.