  Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph,
the vertex cover number is at most $r{-}1$ times the matching number. This
conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$
if the hypergraph is intersecting. There has also been considerable effort made
for finding hypergraphs that are extremal for Ryser's Conjecture, i.e.
$r$-partite hypergraphs whose cover number is $r-1$ times its matching number.
Aside from a few sporadic examples, the set of uniformities $r$ for which
Ryser's Conjecture is known to be tight is limited to those integers for which
a projective plane of order $r-1$ exists.
  We produce a new infinite family of $r$-uniform hypergraphs extremal to
Ryser's Conjecture, which exists whenever a projective plane of order $r-2$
exists. Our construction is flexible enough to produce a large number of
non-isomorphic extremal hypergraphs. In particular, we define what we call the
{\em Ryser poset} of extremal intersecting $r$-partite $r$-uniform hypergraphs
and show that the number of maximal and minimal elements is exponential in
$\sqrt{r}$.
  This provides further evidence for the difficulty of Ryser's Conjecture.
