Cyclic F-manifolds, distinguished connections and integrability

We show that Hertling-Manin F-manifolds provide the appropriate theoretical framework for studying the integrability of quasilinear systems of first-order evolutionary partial differential equations of the form ${\bf u}_t=X\circ {\bf u}_x$ under the mild assumption that $X$ is a cyclic vector field with respect to the F-product $\circ$. This approach is very general and allows us to treat even non-regular systems that were previously beyond the scope of existing techniques. Like in the regular case the information about integrability is contained in a torsionless connection associated with the system and the integrability condition reduces to a geometric condition involving the Riemann tensor of the connection and the structure functions of the product.