The most general way to write flat space metric What is the most general way to write flat space (in d=4 in particular), but still preserving some isometries? In particular I'm interested in the case with 2 isometries, basically by using explicitly only 2/4 coordinates in my metric.
Some examples. We all love to write flat space as:
$$ds_{flat}^2=-dt^2+dx^2+dy^2+dz^2$$
But of course with a change of coordinates to spherical. I can write:
$$ds_{flat}^2=-dt^2+dr^2+r^2 (d\theta^2+\sin^2(\theta)d\phi^2)$$
with the usual range for the coordinates. A less trivial example with oblate spheroidal coordinates:
$$ds_{flat}^2=  -dt^2+{ \rho^2+L^2 \cos^2\vartheta \over \rho^2+L^2}  d\rho^2+  ( \rho^2+L^2 \cos^2\vartheta  ) d\vartheta^2+ (\rho^2+L^2) \sin^2\vartheta  d\varphi^2  $$
for some constant $L$.
The generic ansatz for flat space would be something like $g_{\mu \nu}=g_{\mu \nu}(x_1,x_2)$ in some ${x_1,x_2,x_3,x_4}$ coordinates. References are welcome.
 A: A metric defines a flat space within an open neighborhood $U$ if and only if the Riemann tensor $R$ vanishes over that neighborhood. So you simply have to calculate $R$ and check that it vanishes in the neighborhood. 
The only if part of the assertion is clear. The if part is probably more interesting to you and indeed a simple proof of the if shows you (in principle) how to construct the transformation to co-ordinates that will chart your whole neighborhood with Minkowski co-ordinates. Choose a point within your neighborhood $P$ and a basis $\{e_\mu\}$ for its tangent space. Now we simply use the fact that $R=0$ means that the parallel transport of a basis frame vanishes around a loop; otherwise put, the image under parallel transport of $\{e_\mu\}$ to any other point $Q$ in the neighborhood is independent of the path you tread to get there. So we simply compute this image (in principle) at every point in the neighborhood, and the results will be well defined. In this system, parallel transport of this basis between neighboring points yields the identity transformation, therefore the covariant derivatives $\{\nabla\,e_\mu\}$ all vanish everywhere; a fortiori so do all the commutators $[e_\mu,\,e_\nu]$ and so the vector fields defined by the $e_\mu$ can be integrated to co-ordinates wherein $e_\mu=\frac{\partial}{\partial x^\mu}$, whence the connexion co-oefficients $\Gamma$ must all vanish too, that is, $x^\mu = a^\mu + b^\mu\,\lambda$ define geodesics in these co-ordinates, where $a$ and $b$ are constants and $\lambda$ the path parameter holds good over the whole neighborhood (i.e. it's not simply a local linearization approximating the tangent space to one point).
See Misner Thorne and Wheeler Section 11.5 for the finer details. Also, the present (16th June 2016) version of the Wikipedia article on the Riemann Curvature Tensor explains some of these points well; here's a particularly pithy little phrase from there that is good to remember:

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space

although here, of course, one replaces Euclidean with Minkowskian. Most arguments about and properties of $R$ withstand the change from a Riemannian manifold (wherein the locally diagonalized metrics are nonsignatured) to a Lorentzian one (wherein the diagonalizations have the $(1,\,3)$ signature we all ken and love).
