  The aim of this paper is to give new insights about families of integrable
systems lifting a Hamiltonian $S^1$-space. Specifically, we study one-parameter
families $(M^4,\omega,F_t=(J,H_t))_{0 \leq t \leq 1}$ of systems with a fixed
Hamiltonian $S^1$-space $(M,\omega,J)$ and which are semitoric for certain
values of the parameter $t$, with a focus on such families in which one
singular point undergoes a Hamiltonian-Hopf bifurcation (also called nodal
trade in the context of semitoric systems, and more generally almost toric
fibrations). Beyond semitoric systems, we also study families containing
hypersemitoric systems, and we investigate the local theory of a nodal
trade.\\Building on and generalizing the ideas of a previous paper, we show how
such families can be used to find explicit semitoric systems with certain
desired invariants (bundled in the marked semitoric polygon). This allows us to
make progress on the semitoric minimal model program by understanding and
coming up with explicit systems for each strictly minimal type (i.e., those not
admitting any toric or semitoric type blowdown). In order to obtain these
systems, we develop strategies for constructing and understanding explicit
examples of semitoric (and hypersemitoric) systems in general. One strategy we
make use of is to start from a well-understood system (such as a toric system)
and to explicitly induce Hamiltonian-Hopf bifurcations to produce focus-focus
singular points. This is an expanded version of the technique used in the
aforementioned previous paper, in order to apply it to semitoric systems which
include non-trivial isotropy spheres in the underlying $S^1$-space (i.e.,
$\mathbb{Z}_k$-spheres), which occurs in several of the strictly minimal
systems.\\In particular, we give an explicit one-parameter family of systems on
$\mathbb{CP}^2$ which transitions between being of toric type, semitoric type,
and hypersemitoric type depending on the value of the parameter. We study this
system at each stage, computing the marked semitoric polygon of the semitoric
system and determining several properties of the hypersemitoric system,
including the existence of a unique flap and two parabolic orbits. Furthermore,
we study the transitions between these stages.\\We also come up with new
explicit semitoric systems on all Hirzebruch surfaces which, together with the
previous systems and the systems already contained in the literature, gives an
explicit model for every type of strictly minimal system. Moreover, we show how
to obtain every strictly minimal system by applying sequences of alternating
toric type blowups and blowdowns to simple explicit systems. In particular, we
obtain that every strictly minimal semitoric polygon can be obtained from a
semitoric system which is part of a family $(M,\omega,F_t=(J,H_t))$ which is
semitoric for all but a finite number of values of $t$, called a semitoric
family.
