Answer: $8$ lines.

Fig. 14

Fig. 15

Let $l_1$, $l_3$, $l_5$ be the lines on which $1$, $3$, and $5$ points are marked, respectively, and let $A$ be the only marked point on $l_1$. Then all other lines either pass through this point (call these lines meridians) or are parallel to $l_1$ (call these parallels). Note that each meridian intersects each parallel, and all such intersection points are distinct; moreover, these points exhaust all intersection points except $A$. Thus, each meridian has the same number of marked points, and each parallel as well. From the condition, it follows that either each meridian has $3$ points and each parallel has $5$ (so there are $2$ parallels and $5$ meridians, see Fig. 14), or vice versa (so there are $4$ parallels and $3$ meridians, see Fig. 15). In both cases, there are $8$ lines.