  Given a set $\mathfrak S$ of conformal maps of the unit disk $\mathbb D$ into
itself that is closed under composition, we address the question whether
$\mathfrak S$ can be represented as the reachable set of a Loewner - Kufarev -
type ODE $\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$, $w_0=\mathsf{id}_{\mathbb
D}$, where the control functions $t\mapsto G_t$ form a convex cone $\mathcal
M_{\mathfrak S}$. For the set of all conformal $\varphi:\mathbb D\to\mathbb D$
with $\varphi(0)=0$, $\varphi'(0)>0$, an affirmative answer to this question is
the essence of Loewner's well-known Parametric Representation of univalent
functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of
conformal self-maps defined by their boundary regular fixed points and, in part
of the cases, establish their Loewner-type representability.
