Invariants of sequences for the group $\mathrm{SO}(2,p,\mathbb{Q})$ of a two-dimensional bilinear metric space over the field of rational numbers

Let $\mathbb{Q}$ ne the two-dimensional vector space over the field of rational numbers $\mathbb{Q}$ and $\langle x,y\rangle=x_{1}y_{1}+px_{2}y_{2}$ be a bilinear form on $\mathbb{Q}^{2}$, where $p=1$ or $p=p_{1}\cdot p_{2}\cdot\ldots\cdot p_{n}$; here $p_{j}$ are prime numbers such that $p_{k}\neq p_{l}$ for $k\neq l$, $k\le n$, and $l\le n$. We denote by $\mathrm{O}(2,p,\mathbb{Q})$ the group of all linear transformations of $\mathbb{Q}^{2}$ that preserve the form $\langle x,y\rangle$ and set $\mathrm{SO}(2,p,\mathbb{Q})=\{g\in \mathrm{O}(2,p,\mathbb{Q}): \det(g)=1\}$. This paper is devoted to the problem on the $G$-equivalence of finite sequences of points in $\mathbb{Q}^{2}$ for the group $\mathrm{SO}(2,p,\mathbb{Q})$. We obtain a complete system of $G$-invariants of finite sequences of points in $\mathbb{Q}^{2}$ for the group $G=\mathrm{SO}(2,p,\mathbb{Q})$.