Palindromic pairs in conics

Abstract

For each point $p\in \left( [0,1]\cap \mathbb Q\right) ^2$ we define and classify a family $\left( C(p;k)\right) _{k\in \mathbb Z}$ of rational conics passing through p, each one containing a sequence of rational points $\left( x^k_n,y^k_n\right) _{n\ge -k}$ converging to p and such that the continued fraction of $x^k_n$ and $y^k_n$ are reversal to one another. Moreover the points $\left( x^k_n,y^k_n,k\right) _{k\in \mathbb Z,n\ge -k}$ are contained in a quartic surface Q(p) whose intersection with horizontal planes are conics.