On diagonal cubic surfaces

We show that the affine surfacesx³+y³+cz³=c, c Q, in the casesc2,c=2, contain precisely 2, respectively 4, polynomial parametric solutions corresponding to curves of arithmetic genus 0 on the surface.However, these surfaces contain infinitely many polynomial parametric solutions corresponding to curves of arithmetic genus greater than 0.The author wishes to acknowledge the receipt of a Summer Support Grant for the College of Liberal Arts, Arizona State University, while this note was being written.