  Corrado B\"ohm once observed that if $Y$ is any fixed point combinator (fpc),
then $Y(\lambda yx.x(yx))$ is again fpc. He thus discovered the first "fpc
generating scheme" -- a generic way to build new fpcs from old. Continuing this
idea, define an $\textit{fpc generator}$ to be any sequence of terms
$G_1,\dots,G_n$ such that \[ Y \in FPC \Rightarrow Y G_1 \cdots G_n \in FPC \]
In this contribution, we take first steps in studying the structure of (weak)
fpc generators. We isolate several robust classes of such generators, by
examining their elementary properties like injectivity and (weak) constancy. We
provide sufficient conditions for existence of fixed points of a given
generator $(G_1,\cdots,G_n)$: an fpc $Y$ such that $Y = Y G_1 \cdots G_n$. We
conjecture that weak constancy is a necessary condition for existence of such
(higher-order) fixed points. This statement generalizes Statman's conjecture on
non-existence of "double fpcs": fixed points of the generator $(G) = (\lambda
yx.x(yx))$ discovered by B\"ohm.
  Finally, we define and make a few observations about the monoid of (weak) fpc
generators. This enables us to formulate new a conjecture about their
structure.
