= Solution
An orientation-preserving isometry fixing every point of $\ell_+$ fixes more than two points in the upper half-plane and hence, as a <Möbius transformation>, is the identity. The map
$$
R(z)=-\overline z
$$
fixes $\ell_+$ pointwise and preserves the hyperbolic metric; it is the <hyperbolic reflection> in $\ell_+$. Consequently the pointwise stabilizer is exactly
$$
\boxed{\{\operatorname{id},R\}}.
$$
For any isometry $F$, use part (a) to choose $g\in PSL_2(\mathbb R)$ that agrees with $F$ on one pointed line and its tangent direction. Then $g^{-1}F$ fixes that line pointwise, so it is either the identity or its reflection. Hence the <Isometry group of the Poincare half-plane> consists of
$$
z\longmapsto\frac{az+b}{cz+d}
\quad(ad-bc>0)
$$
and
$$
z\longmapsto\frac{a\overline z+b}{c\overline z+d}
\quad(ad-bc<0),
$$
with real coefficients, modulo multiplication of all four coefficients by a nonzero scalar.
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