Many problems in Applied Mathematics lead to the study of the nonlinear partial differential equation \(u_ t = (a(u))_{xx} + (b(u))_ x + c(u)\). The interest in the existence of travelling-wave solutions of the form \(U(x,t) = U(\psi)\), \(\psi = x - \lambda t\), originates an ordinary differential equation from which arise integral equations of the form \((*)\) \(y(t) = f(t) + \int^ t_ 0 g(s)/y(s)ds\), where \(y\) is the unknown. \((*)\) is a nonlinear Voterra integral equation of the second kind with singular integrand. In the paper, the author studies the existence of nonnegative continuous solutions of the equation \((*)\), showing that it has no solutions, a unique solution or an uncountable number of solutions. In this last case, he proves that there is one which is maximal, providing a method for constructing it. From this, he establishes necessary and sufficient conditions for the existence of solutions.