Small Solutions of generic ternary quadratic congruences

We consider small solutions of quadratic congruences of the form x₁²+₂x₂²+₃x₃² 0 q, where q=pᵐ is an odd prime power. Here, ₂ is arbitrary but fixed and ₃ is variable, and we assume that (₂₃, p) =1. We show that for all ₃ modulo pᵐ which are coprime to p except for a small number of ₃'s, an asymptotic formula for the number of solutions (x₁, x₂, x₃) to the congruence x₁²+₂x₂²+₃x₃² 0 q with \|x₁|, |x₂|, |x₃|\ N holds if N q^11/24+ and q is large enough. It is of significance that we break the barrier 1/2 in the above exponent. If q is restricted to powers of a fixed prime p, we obtain a slight improvement of this result using the theory of p-adic exponent pairs, as developed by Mili\'cevi\'c, replacing the exponent 11/24 above by 11/25. Under the Lindel\"of hypothesis for Dirichlet L-functions, we are able to replace the exponent 11/24 above by 1/3.