Two-parameter families of uniquely extendable Diophantine triples

Let A and K be positive integers and ε∈ {-2,-1,1,2}. The main contribution of the paper is a proof that each of the D（ε-2）-triples {K, A-2 K＋2εA,（A ＋1）-2 K ＋ 2ε（A＋1）} has uniqui extension to a D（ε^2）-quadruple. This is used to slightly strengthen the conditions required for the existencc of a D（1）-quintuple whose smallest three elements form a regular triple.