The equation of motion for a bead in a chain of uncompressed elastic beads in contact that interact via the potential V(delta) approximately delta( n), n>2, delta being overlap, supports solitary waves and does not accommodate sound propagation [V. Nesterenko, J. Appl. Mech. Tech. Phys. 5, 733 (1983)]. We present an iteratively exact solution to describe the solitary wave as a function of material parameters and a universal, infinite set of coefficients, which depend only on n. We compute any arbitrary number of coefficients to desired accuracy and show that only the first few coefficients of our solution significantly improves upon Nesterenko's solution. The improved solution is a necessary step to develop a theoretical understanding of the formation of secondary solitary waves [M. Manciu, et al., Phys. Rev. E 63, 011614 (2001)].