Dear Jeff,
I hear your objections about the Red Cross, but it by far and away would be the leading candidate. The Cross (versus the Red Crescent) is the logo in a number of Islamic nations (Indonesia, Mali, Kosovo) and throughout non-Christian eastern Asia. If the issue is "real-world 2019" issues (ie. YOU cannot publish that logo, even for use in a fictional setting), then just change the colour. In general terms:
* white cross on green = first aid (per International Standards Organisation), or occupational health
* green cross on white = alternative symbol for first aid (per ISO), but used by a few private companies
* white cross on blue = used to indicate a hospital, usually on road maps or street signs (perhaps this is just an Australian thing)
You get the idea. If colour is disassociated from the symbol (as per the Imperial Sunburst), then there is no reason to NOT have the cross as the symbol.
[BTW, to separate the symbol from Christian symbols, a difference needs to be drawn between the "Greek cross" - four equally long arms - and the Red Cross - arms with length equal to their width. In other words, the Red Cross is stubbier than any previous Christian cross variant.]
Alternatives?
The snake +/- staff of Asclepius (For the love of God, NOT the caduceus ... that symbol of the god of merchants and thieves!)
The Star of Life (A bit US-centric)
An Ankh (especially if simplified to a "crux ansata" - a melding of the Red Cross and a classic ankh)
Non-Solomani:
* The Star of David, perhaps stretched in the Y-axis (Bear with me here ...)
Separate to its Hebrew connotations, I can see the same symbol as overlap of a gynoid figure (upward pointing triangle) and an android figure (downward pointing triangle) and, thus, a symbol for the totality of humanity. (Much like overlapping Mars/Venus icons.)
* A letter "T", perhaps with arms running slightly downwards.
Human figure, arrow point up, aspiration to raise people to higher things. Perhaps stylised shugilii blessing food.
Cannot see a separate Sylean symbol. At the time of Vilani contact in -9000 or so, the Syleans were pre-industrial. Any native Sylean symbology is likely to be overwhelmed by Vilani signage and design technology standards. By the time of Terran conquest 6500 years later, the population of Sylea was 99.5% Vilani (see GURPS: Humaniti, p. 129). Then add a few hundred years of Terran symbology before any Sylean contacts are limited to their own pocket empire for another few hundred years. Is that long enough for an entirely new set of symbology to spring up? (Consider how ancient the proposed Solomani symbols are!) It would be kind of like the modern USA being cut off from the rest of the world from 200 years and, on re-contact, found to be using stylised Native American symbols (or any other radially new set of symbols) rather than the modern (actually quite ancient) icons we are familiar with.
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Cheers!
Ken
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Kenneth Barns
Mobile: +61 4 5957 2825 / 04 5957 2825