Solutions of x12 + x22- x32 = n2 with small x3

Friedlander and Iwaniec investigated integral solutions (x(1), x(2), x(3)) of the equation x(1)(2) + x(2)(2) - x(3)(2) = D, where D is square-free and satisfies the congruence condition D = 5 mod 8. They obtained an asymptotic formula for solutions with x(3) = M, where M is much smaller than root D. To be precise, their condition is M >= D1/2-1/1332. Their analysis led them to averages of certainWeyl sums. The condition of D being squarefree is essential in their work. We investigate the "opposite" case when D = n(2) is a square of an odd integer n. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with x(3) = M, where M >= D1/2-1/16+epsilon.