Small solutions of ternary quadratic congruences with averaging over the moduli

In a recent paper, we proved that for any large enough odd modulus $q\in \mathbb{N}$ and fixed ${\alpha }_2\in \mathbb{N}$ coprime to $q$, the congruence $$x_1^2+{\alpha }_2{x}_2^2+{\alpha }_3{x}_3^2\equiv 0q$$ has a solution of $(x_1,x_2,x_3)\in {\mathbb{Z}}^3$ with $x_3$ coprime to $q$ of height $\max \{\mathrm{\mid }{x}_1\mathrm{\mid },\mathrm{\mid }{x}_2\mathrm{\mid },\mathrm{\mid }{x}_3\mathrm{\mid }\}\le q^{11\mathrm{/}24+\epsilon }$ for, in a sense, almost all ${\alpha }_3$, where ${\alpha }_3$ runs over the reduced residue classes modulo $q$. Here it was of significance that $11\mathrm{/}24<1\mathrm{/}2$, so we broke a natural barrier. In this paper, we average the moduli $q$ in addition, establishing the existence of a solution of height $\le Q^{3\mathrm{/}8+\epsilon }{\alpha }_2^{\epsilon }$ for almost all pairs $(q,{\alpha }_3)$, with $Q$ large enough, $Q<q\le 2Q$, $q$ coprime to $2{\alpha }_2$ and ${\alpha }_3$ running over the reduced residue classes modulo $q$.