In this paper we continue with the problem of the existence of entire solutionsFandGto the functional equationwherePandQare given polynomials andHis a polynomial Typically, the results are that, except for certain exceptional cases, the only entire solutionsFandGare polynomial. Often, one can then show, for a particular given pair of polynomialsPandQ, that no polynomial solutionsFandGmay exist. This problem is closely related to the problem of extending harmonically to the entire plane the solution to a Dirichlet problem on an algebraic curve with polynomial boundary data. This paper is closely related to earlier work of Flatto, Newman and Shapiro, Baker and Gross, Flatto, and the present authors.