From reading the log entry about the painting (The Tunnel of Communication) as well as certain other minor clues, you know the parabolic mirror was used as the earth / moon communication system. You need to repair the broken lumen machine first as it provides power to the mirror and produces lumen, which you need later.


One of the few bits of free information Scrupul gave was that azotic acid and blue ore are used to make lumen, so you need to fetch these items, remembering to first add another yellow and green compote to the food vat.
The broken lumen machine on the first level is in pretty bad shape and needs extensive repairs. The easiest way to go about this is to begin by attaching the handle. Examine all the hotspots noting that the control box on the far right has an indentation in the shape of the handle. Fit the handle and then pull it. Although the machine won't work, the screen automatically pans to the next part to be fitted or repaired. Repeat this process until all the repairs have been done.
Two parts are missing from the bottom left of the machine. The cogged cylinder resembles the crushing rollers, but it's incomplete. By definition a cylinder is hollow and the ax (right size and shape) will fit. Crushing rollers also have a cog (or gear) at the end, so you need another coloured gear from the device at the top of the mountain. Combine the three items to make a crushing roller that fits into the machine, between the two existing ones.


Attach the hopper to the machine (below the crushing rollers) and fill it with the blue ore. Pull the handle again and the screen pans to the fuel line. Insert the funnel in the fuel line to prevent spillage, then pour in the azotic acid. The holes in the pipes are plugged with glue and the nipple at the top must be greased. The machine should now work when you pull handle, producing lumen in the receptacle on the right. In addition, the pneumatic press in the middle can be used to straighten the twisted plating, restoring it to a protective shield.


The broken mirror you found in the Hall of Secret Antiquities can be fixed with glue, then used to replace the smashed one on the parabolic mirror. Place the control box (from the gilded figures puzzle) in the centre of the three pipes, then start the lumen machine to provide power. The fresco on the wall in the Hall of Secret Antiquities depicted a parabolic mirror and two ideograms (light and fire) with numbers below them.

The same two ideograms, albeit in reverse order, are also on the control box. Set the indicators to the numbers on the fresco (7 and 4) by using the directional arrows at the bottom.
Ardan is able to communicate with earth using Morse code generated by the mirror. All the items offered, except the earth plants, are freely available on the moon and there's no reason to request anything. However, the earth plants are required to solve an optional extra puzzle in the game, so you can request them. Take note of Ardan's thoughts about the shell crashing in the crater, so make your way there to collect it. From either the fifth or bottom ledge look closely at the forests and smoke will be seen rising from the one in which the shell landed. The explosion of the shell scatters the earth plants and one of them has hybridized with a local plant, producing lunar cotton.
Note that for other items requested a canister (box) containing the item will be found intact. You need your can opener to open the box, so it must first be repaired with some glue.
You can request any two items from earth (click the control box for the second request) after which the space cannon on earth is damaged and no further shells can be launched.


From the log entry about the painting you also learn that there appears to be another painting underneath, that can be revealed using a moistened pad and dissolvent. Get some more Essence of Klipsaggt (it need not be three measures) from the Hall of Secret Antiquities, then combine it with some lunar cotton and the existing painting. This replaces the existing painting with the one underneath. Read the log entry for the new painting, which contains another set of clues to the organ water supply puzzle. This painting, Lumenplay in a Grotto, depicts 'one of the rooms dug in a huge black rock sculpted to resemble the first Supreme Lunar Ruler, located in the middle of the crater'. There's a reference to an inscription: 'How many times in the space of one lunar day and one lunar night does the earth hide it's face? Place the twenties near the right eye, the digits near the left eye, and you will attain the treasure hidden in the nasal cavities of the Most High'.
The first clue is pretty straightforward, but the second one requires some explanation. Put another way the question is referring to the number of earth days there are in a lunar day. The answer is approximately 28. In case any astronomy buffs are reading this, the actual answer is 27,32. Converted into the Selenite number system, 28 is depicted with 'one dot' (1 x 20) in the left column and 'three dots above a bar' (8) in the right column. Note that although the diagram has the twenties on the left, this is actually the skull's right eye. The same applies to the digits.


Phew! All this and it's not even required to complete the game!