  For some time the Petersen graph has been the only known Snark with circular
flow number $5$ (or more, as long as the assertion of Tutte's $5$-flow
Conjecture is in doubt). Although infinitely many such snarks were presented
eight years ago by Macajova and Raspaud, the variety of known methods to
construct them and the structure of the obtained graphs were still rather
limited. We start this article with an analysis of sets of flow values, which
can be transferred through flow networks with the flow on each edge restricted
to the open interval $(1,4)$ modulo $5$. All these sets are symmetric unions of
open integer intervals in the ring $\mathbb{R}/5\mathbb{Z}$. We use the results
to design an arsenal of methods for constructing snarks $S$ with circular flow
number $\phi_c(S)\ge 5$. As one indication to the diversity and density of the
obtained family of graphs, we show that it is sufficiently rich so that the
corresponding recognition problem is NP-complete.
