Equidistribution of solutions of ternary quadratic congruences modulo prime powers

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