Two sets with an infinite number of new systems of orthogonal polynomials have recently been discovered by Smith (1982) in connection with some nonlinear physical problems, e.g. the dispersion of a buoyant contaminant in a fluid. They appear as solutions of nonlinear differential equations. Let (Pn(x; m, k, delta )) and (Qn(x; m,k)), with n=0, k, m+k, 2m, 2m+k,. . ., denote a generic system of each set. The positive integers k and m are restricted by k(m and delta )1-k. Although the orthogonality interval of these polynomials is real, their zeros are generally complex. Here the sum rules yr= Sigma xri,n, r=1,2,. . ., for the zeros (xi,n; i=1,2,. . ., n) of the n-th degree polynomials of these sets are studied. It is found that all these quantities vanish except for r=pm, p being an arbitrary positive integer. Simple recurrent expressions for ypm are given.