updates made mainly in the introduction section and i would like to say more but my knowledge of the historical accounts on KK theory ran short besides, my plan was to make this draft as short as possible. the draft is still in the process of completion…
https://drive.google.com/file/d/1si4ifALrdrQUOaN9AUknjVBf_lMtV7Ao/view?usp=sharing

static case is now tackled
https://foxtrotpaparomeo.wordpress.com/2023/09/07/07092023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/

in the static setting, it's straightforward to establish correspondence to Schwarzschild solution yet, there is also the temptation to consider a setting where the fields evolve with both lower dimensional coordinates

in this (1+1)-dimensionally reduced form of GR, the form of the conformal field is handed down by an auxiliary field not from the equation of motion for the conformal field. this auxiliary field is put in by hand and the acceptable form must satisfy a constraint resulting from the equation of motion for the conformal field.

in this: the equation of motion for the conformal scalar field does not solve for a particular form of this scalar, instead this equation of motion yields an identity equation... otoh, this conformal scalar is a functional of an auxiliary scalar field which itself does not have a specific form that is the only acceptable, not just one form can satisfy the identity equation
https://foxtrotpaparomeo.wordpress.com/2023/11/27/27112023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-dimensions/

updated it today, you may go directly to (22.11) for the de Sitter-like solution
https://foxtrotpaparomeo.wordpress.com/2023/12/10/10122023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/

additional basic resource material to my venture into 1 + 1 GR (you can refer to eqn (23) --- the vanishing of Einstein's tensor strictly in 1 + 1 dim)
https://digitalcommons.calpoly.edu/physsp/65/

(additional resource material re G -- measuring G during the big bang nucleosynthesis and comparing it to its present value, results at 2sigma)
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When combining the measured abundances and the baryon density from CMB observations by Planck, we find G_BBN/G_0 = 0.99^+0.06_−0.05 at 2σ confidence level.
https://arxiv.org/abs/1910.10730