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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 systemsfor large values of the parameter $\alpha$. We determine the top critical valueof $\alpha$ 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 thesame as for large $\alpha$. We give some detailed results for $g\le5$ andapplications to higher rank Brill-Noether theory and the stability of kernelsof evaluation maps, thus proving Butler's conjecture in some cases in which itwas not previously known.

On coherent systems of type (n,d,n+1) on Petri curves