Equidistribution of Solutions of Ternary Quadratic Congruences Modulo Prime Powers

Let $p$ be a fixed odd prime and $Q(x,y,z)=a{x}^2+bxy+c{y}^2+dxz+eyz+f{z}^2$ be a fixed quadratic form in $\mathbb{Z}[x,y,z]$ which is non-degenerate in ${\mathbb{F}}_p[x,y,z]$ and $(a(4ac-b^2),p)=1.$ Let $(x_0,y_0,z_0)$ be a fixed point in ${\mathbb{Z}}^3$. We study the behavior of solutions $(x,y,z)$ of congruences of the form $Q(x,y,z)\equiv 0q$ with $q=p^n,$ where max$\{\mathrm{\mid }x-x_0\mathrm{\mid },\mathrm{\mid }y-y_0\mathrm{\mid },\mathrm{\mid }z-z_0\mathrm{\mid }\}\le N$ and $(z,p)=1.$ In fact, we consider a smooth version of this problem and establish an asymptotic formula (thus the existence of such solutions) when $n\to \mathrm{\infty }$, under the condition $N\ge q^{\frac{1}{2}+\epsilon }$.