Axially symmetric metric Can an axially symmetric metric depend on the coordinate $\phi$?
Usually it is said that a metric that does not depend on the coordinate $\phi$ is axially symmetric since the vector $$\xi=\nabla\phi=(0,0,0,1)$$ is a Killing vector, but are there examples of  metrics which depend on $\phi$ and still have $\xi$ as a Killing vector?
 A: Spacetime doesn't come equipped with coordinates, so there is no reason to think that we automatically have anything called $\phi$ that works like an azimuthal angle, nor is there any guarantee that we can construct such a coordinate. This makes the problem different from the problem of defining something like the axial symmetry of a mass distribution in flat 3-space.
A coordinate-free definition of axial symmetry is the following. A spacetime S is axially symmetric if it can be written as a union $\text{S}=\cup c_{z,t}$ where each of the $c$ varies smoothly with $z$ and $t$ and is a cylinder, meaning that it's spacelike, has zero intrinsic curvature, and has the topology of a cylinder.

are there examples of metrics which depend on  and still have  as a Killing vector?

If you want to say this in terms of Killing vectors, then you can change the description to one that says each $c$ has Killing vectors that act algebraically like those of a cylinder. That is, it has two Killing vectors, these Killing vectors commute (unlike those of a sphere), and one of them is a field with closed orbits.
A: The article Axially symmetric spacetimes: numerical and analytical perspectives is a very nice reference and treats axially symmetric spacetimes in generality, showing how to construct the "cylindrical" slices (see also this).
If your spacetime is not only  "axially symmetric" but also "circular" (i.e. there are no meridional flows of matter), then there is a natural way to construct the coordinate $\phi$ ("circular" = there is a spacelike Killing vector field $\xi$ that has closed orbits and vanishes on a timelike worldsheet).
Such a construction is described in these slides of John Friedman. More details are given in his book.
A freely downloadable resource are these extremely good lecture notes: take a at look section 2.
PS: given the construction of $\phi$ as $\xi = \nabla \phi$, the metric does not depend on $\phi$ when $\phi$ is taken as coordinate.
