  Let $k$ be an infinite field of characteristic 0, and $X$ a del Pezzo surface
of degree $d$ with at least one $k$-rational point. Various methods from
algebraic geometry and arithmetic statistics have shown the Zariski density of
the set $X(k)$ of $k$-rational points in $X$ for $d\geq2$ (under an extra
condition for $d=2$), but fail to work in generality when the degree of $X$ is
1, leaving a large class of del Pezzo surfaces for which the question of
density of rational points is still open. In this paper, we prove the Zariski
density of $X(k)$ when $X$ has degree 1 and is represented in the weighted
projective space $\mathbb{P}(2,3,1,1)$ with coordinates $x,y,z,w$ by an
equation of the form $y^2=x^3+az^6+bz^3w^3+cw^6$ for $a,b,c\in k$ with $a,c$
non-zero, under the condition that the elliptic surface obtained by blowing up
the base point of the anticanonical linear system $|-K_X|$ contains a smooth
fiber above a point in $\mathbb{P}^1\setminus\{(1:0),(0:1)\}$ with positive
rank over $k$. When $k$ is of finite type over $\mathbb{Q}$, this condition is
sufficient and necessary.
