Cardano triplet

A triple of positive integers $(a,b,c)$ satisfying $\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=1$, with real cube roots. Cubing the equality yields the equivalent integer identity $27b^{2}c=(a+1{)}^2(8a-1)$. In particular $a=3k-1$ for a positive integer $k$, and the condition becomes $b^{2}c=k^2(8k-3)$.