Classification of homogeneous locally nilpotent derivations of k[X,Y,Z] Part I: Positive gradings of positive type

Let k be a field of characteristic zero, let a,b,c be relatively prime positive integers, and define a grading "g' on the polynomial ring B = kX,Y,Z] by declaring that X,Y,Z are homogeneous of degrees a,b,c, respectively. Consider the problem of classifying g-homogeneous locally nilpotent derivations of B. The present paper solves the case where g has positive type, which means that a,b,c are not pairwise relatively prime. The case where a,b,c are pairwise relatively prime is solved in our subsequent paper.