  We show that F-manifolds with compatible connection $(\nabla,\circ,e)$ are
the geometric counterpart of integrable systems of quasilinear first order
evolutionary PDEs. We consider F-manifolds equipped with an Euler vector field
and assume that the operator $L=E\circ$ is regular. This generalises previous
results in the semisimple context. As an example we study regular F-manifolds
with compatible connection $(\nabla,\circ,e,E)$ associated with integrable
hierarchies obtained from the solutions of the equation $d\cdot d_L \,a_0=0$ by
applying the construction of [27]. We show that $n$-dimensional F-manifolds
associated to operators $L$ with $r\le n$ Jordan blocks $L_\alpha$ of size
$m_\alpha$ are classified by $n$ arbitrary functions of a single variable,
where each block $L_\alpha$ contributes with $m_\alpha$ functions of the
variable appearing in the diagonal of the block. In the case of a single Jordan
block of arbitrary size we show that flat connections $\nabla$ correspond to
linear solutions $a_0$. This generalises part of the construction of [31] where
special linear solutions were considered. We illustrate the construction in
dimensions $2,3,$ and $4$ for any choice of Jordan canonical form and any
choice of the corresponding solution $a_0$. In these dimensions we have that
linear solutions define bi-flat F-manifolds, and that the special linear
solutions studied in [31] are related to Riemannian F-manifolds with Killing
unit vector field. We conjecture that this is true in general.
