Let each line of a projective plane $\ P\ $ have $\ p\ge 5\ $ points. Let
lines $\ L_0\,\ L_1\,\ L_2\ $ of $\ P\ $ have empty intersection
$\ L_0\cap L_1\cap L_2\ =\ \emptyset.\ $ Define

$$ H\ :=\ P\setminus(L_0\cup L_1\cup L_2) $$

The H-lines are defined as sets $\ H\cap L,\ $ where $\ L\ $ is a projective line in $\ P.$

Then $\ H,\ $ together with H-lines, is a hyperbolic plane such that each H-line has $\ p-2\ $ or $\ p-3\ $ points, where both these cardinalities do happen.

>Let me present a proof of property 4: **INDEED**, let $\ x\in H\setminus L,\ $ where L is an H-line, and let $\ L'\ $ be the projective extension of $\ L.\ $ For each $\ y\in L'\cap M',\ $ where $\ M'\ $ is one of the removed projective lines, we get the H-line $\ xy\cap H\ $ which is parallel to $\ L,\ $ where $\ xy\ $ is the projective line passing through $\ x\ $ and $\ y.$
>
>There is also a removed line $\ N'\ $ which doesn't pass through $\ y\ $ (the intersection of all removed lines is assumed to be empty). Thus $\ \exists_z\ y\ne z\in L'\cap N',\ $ and $\ xz\cap H\ $ is another H-line parallel to $\ L\ $ different from the previous parallel line.

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More generally, one may consider any family of three or more projective lines (instead of three of them), which have an empty intersection. Then, with a bit of care, one gets (infinitely) more required examples with more than two different cardinalities of the H-lines.