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 $$C_{a,b}:x^4+y^4+z^4+a{x}^2{y}^2+b(x^2+y^2)z^2=0,$$ such that $(a^2-4)(b^2-4)(a^2-1)(b^2-a-2)\mathrm{\ne }0.$ Furthermore, the geometry of these points is investigated.