  We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus
$g\ge2$. We describe the geometry of the moduli space of such coherent systems
for large values of the parameter $\alpha$. We determine the top critical value
of $\alpha$ and show that the corresponding ``flip'' has positive codimension.
We investigate also the non-emptiness of the moduli space for smaller values of
$\alpha$, proving in many cases that the condition for non-emptiness is the
same as for large $\alpha$. We give some detailed results for $g\le5$ and
applications to higher rank Brill-Noether theory and the stability of kernels
of evaluation maps, thus proving Butler's conjecture in some cases in which it
was not previously known.
