This paper deals with the problem of determining the leading coefficient k = k( (u')(2)) of the nonlinear (monotone potential) Sturm-Liouville operator Au = - (k ((u') (2)) u' (x))' + q(x)u(x), x epsilon (a,b). As an additional condition only two measured data at the boundary (x = a, x = b) are used. Solvability and linearization of the corresponding nonlinear direct problein are given. An existence of a quasi-solution of the inverse problem is obtained in a suitable compact class of admissible coefficients. In the second part of the paper an approximate analytical solution for the inverse problem is derived. The approach presented permits to analyze well-posed, as well as, all ill-posed situations for the inverse coefficient problem. Numerical examples corresponding to the Lill considered situations are presented. (C) 2003 Elsevier Inc. All rights reserved.