Cox rings of K3 surfaces of Picard number three

Let X be a projective K3 surface over C. We prove that its Cox ring has generating set whose degrees are either classes of smooth rational curves, sums at most three elements the Hilbert basis nef cone, or form 2(f+f?), where f,f? elliptic curves with f?f?=2. This result and techniques using Koszul's type exact sequences then applied to determine for all Mori dream surfaces Picard number which is minimal in cases. A presentation given some special cases few generators.