Conformastatic spacetimes

Conformastatic spacetimes refer to a special class of static solutions to Einstein's equation in general relativity.
Introduction.
The line element for the conformastatic class of solutions in Weyl's canonical coordinates reads
formula_1
as a solution to the field equation
formula_2
Eq(1) has only one metric function formula_3 to be identified, and for each concrete formula_3, Eq(1) would yields a "specific" conformastatic spacetime.
Reduced electrovac field equations.
In consistency with the conformastatic geometry Eq(1), the electrostatic field would arise from an electrostatic potential formula_5 without spatial symmetry:
formula_6
which would yield the electromagnetic field tensor formula_7 by
formula_8
as well as the corresponding stress–energy tensor by 
formula_9
Plug Eq(1) and Eqs(3)(4)(5) into "trace-free" (R=0) Einstein's field equation, and one could obtain the reduced field equations for the metric function formula_3:
formula_11
formula_12
where formula_13 and formula_14 are respectively the generic Laplace and gradient operators. in Eq(7), formula_15 run freely over the coordinates formula_16.
Examples.
Extremal Reissner–Nordström spacetime.
The extremal Reissner–Nordström spacetime is a typical conformastatic solution. In this case, the metric function is identified as
formula_17
which put Eq(1) into the concrete form
formula_18
Applying the transformations
formula_19
one obtains the usual form of the line element of extremal Reissner–Nordström solution,
formula_20
Charged dust disks.
Some conformastatic solutions have been adopted to describe charged dust disks.
Comparison with Weyl spacetimes.
Many solutions, such as the extremal Reissner–Nordström solution discussed above, can be treated as either a conformastatic metric or Weyl metric, so it would be helpful to make a comparison between them. The Weyl spacetimes refer to the static, axisymmetric class of solutions to Einstein's equation, whose line element takes the following form (still in Weyl's canonical coordinates): 
formula_21
Hence, a Weyl solution become conformastatic if the metric function formula_22 vanishes, and the other metric function formula_23 drops the axial symmetry:
formula_24
The Weyl electrovac field equations would reduce to the following ones with formula_22:
formula_26 
formula_27 
formula_28 
formula_29 
formula_30
where formula_31 and formula_32 are respectively the reduced "cylindrically symmetric" Laplace and gradient operators.
It is also noticeable that, Eqs(14) for Weyl are "consistent but not identical" with the conformastatic Eqs(6)(7) above.