Classification of 1-Weierstrass points on Kuribayashi quartics, I (with two parameters)

In this paper, we classify the 1-Weierstrass points of the Kuribayashi quartic curves with two parameters $a$ and $b$ defined by the equation $$C_{a,b}:x^{4}+y^{4}+z^{4}+ax^{2}y^{2}+b(x^{2}+y^{2})z^{2}=0,$$ such that $(a^2-4)(b^2-4)(a^2-1)(b^{2}-a-2)\neq0.$ Furthermore, the geometry of these points is investigated.