Per Page 31 of TC&TC, the Suhhar Suleyman's magic is much looser than the Order of Hermes', and while a majority of sahirs use Sihr, not all do, and Solomonic magic and naranjs can be created from any "summoning art." However, like with most hedge-wizards, the rules for their creation are somewhat loose and left to the troupe to figure out. Accordingly, as I have way too much time on my hands and I happen to like the flavor of non-Hermetic traditions, I came up with a bit of homebrew for several variants of Sahir.
Principles
Solomonic magic does not have a Magic Theory, but it is clearly a heavily integrated form of magic like the Hermetic Arts. In no case is this more true than with the Circle, and nearly any summoning magic that can target spirits can be integrated into Solomonic magic. By 1220, most of the spirit-summoning Arts known to the Suhhar have been integrated with the Solomonic Arts and can be used as Summoning Arts. A new Summoning Art would require a Minor Breakthrough before it could be used to create naranjs; the most obvious example would be the Rego Technique (though it would be far more likely that a Hermetic magus would instead create the Hermetic True Names virtue and use formulaic Hermetic spells, instead of trying to copy the Solomonic naranj). Any summoner who has learned at least one Solomonic Art, or who has studied under someone who knows naranjs, may learn to create naranjs for himself.
A naranj is single-Art. Some Traditions normally use a combination of Abilities for their summoning magic, but except as otherwise noted, the summoning spell uses only one of the relevant Arts. However, if the Summoning Art would require another Art to function, the sahir must have that Art.
Exempla Gratia: An elementalist with (Theurgical) Summoning 10 and Elementalist Air 4 binds an air spirit into a naranj. The Summoning Total will be based on his score in Theurgical Summoning, which is 10, and not on Summoning+Elementalist Air, which is 14. However, the elementalist could not cast a naranj to summon a fire spirit, unless he also had Elementalist Fire.
...But will they come when you call? Unless otherwise noted by the ability in question, a spirit is not compelled to obey or bargain with its summoner (another advantage of Sihr), though many will do so anyway. The bargaining rules on pg. 47 of TC&TC are only for Sihr and jinn; other Methods either have their own rules for bargaining, or simply require agreement on both sides.
Design Note: Where possible, I'm trying to ensure that Sihr is the most broadly powerful and useful form of Solomonic summoning, which a Rego Vim naranj or even a Theurgical Summoning+Element naranj would break through greater Summoning Strength.
I assume that a Gifted standard sahir does not need to buy Sihr as a Virtue, though a literal interpretation of the RAW would suggest that they do (screw the RAW, in this case). Likewise, other sorts of Gifted sahir should get their Summoning Arts free, though the exact mechanism depends on the tradition.
Finally, I'm flexing the rule about how Supernatural Powers always require a Virtue in the case of Infernal and Divine Powers, because Infernal Powers are supposed to be easy to learn if you can find a teacher and don't mind losing your soul, and Divine Powers are just as easy if you have True Faith. Accordingly, I'm creating two Major Virtues for Infernal Tradition and Divine Tradition, which allow you to freely pick up the four Supernatural Abilities associated with your tradition's magic (the latter requires True Faith). This is a kludge, because the rules as they stand are very poor and create characters who are weaker than they should be. (I ought to make a better rule for these someday...)
On to the sahir traditions!
Elementalist Sahirs
Summoning Art: (Theurgical) Summoning
Warping: Elemental Warping or Fatra Bayna; rarely, Divine Ascent.
Elementalists use Theurgical Summoning to call spirits associated with the elemental Forms and bind them into bottles. Elementalism is relatively common throughout the Middle East, though theurgical elementalism is somewhat less common than the other forms. Theurgical summoners are prized at the buyut al-hikma for their ability to bind elemental spirits into items, and to summon elementals to mine them for vis.
While there are many elementalist traditions, the Apostles of Apollonius are a special case due to their focus on the Divine. Many believe that the Apollonians have developed a Divine variant of the theurgical arts, and in any case, they often have less difficulty than other Solomonic magi in working their arts within the boundaries of the Dominion due to their practice of the Ars Notoria, also an Apollonian work.
Character Creation: UnGifted elementalists are normally Mythic Companions, learning Theurgical Summoning, Theurgical Controlling and the Elementalist Art corresponding to their favored humor, as well as whatever Solomonic Art they favor. Gifted Elementalist-Sahirs are a special case. With troupe permission, they may begin with a double-opened Gift, learning Theurgical Summoning, Theurgical Controlling and the Elementalist Arts as their "summoning art;" otherwise, it is recommended that they open the Solomonic Arts, learn Summoning as their "free summoning art" and take Controlling and their Elementalist Art. It is also possible to have a Companion-level elementalist who doesn't have Controlling, and who either bargains with spirits (perhaps a Spirit Companion) or summons and traps elementals for other sahirs to use or to refine into vis.
Whether Apollonians are allowed Divine elementalism or not is a troupe decision; if allowed, it should be a Major Virtue and, of course, mean that the Solomonic Arts cannot be used in a sinful manner. (Holy Solomonic Magic is also possible, use the Hermetic rules for Holy Magic for inspiration if you want to allow this.) Another option is for Gifted Apollonians to use the Ars Notoria virtue from RoP:D (also a creation of Apollonius) to reduce penalties for casting in a divine aura, but this is a somewhat impractical expenditure of XP and is very heavy on the Virtues.
Mechanical Notes: Elementalists can use Solomonic Storytelling for seeking and scouring like any other sahir, and likewise create naranjs. They can't bind anything but elementals into a container, though; a naranj or a use of Controlling must penetrate Magic Resistance (you did learn that jinni's True Name, didn't you?).
Goetic Summoners
Summoning Art: (Goetic) Summoning
Warping: Vituperation for infernalists, Fatra Bayna for purely Magical or Faerie summoners.
Summoners use the Ars Goetia to summon incorporeal spirits for their own ends. The Ars Goetia is always tainted with the substance of the Infernal, of course, and the higher Goetic Arts always draw on that Realm's power, so the Suhhar is generally somewhat suspicious of goetic summoners; however, as the Suhhar does not share the Order of Hermes' law against infernalism, a Summoner of this type may be found living and working among sahirs. The greatest of these summoners in the Mythic Middle East are the Mulhidun, but very few Mulhidun have joined the Suhhar or learned Solomonic magic, preferring instead to delve deeper into their own dark magic.
Some lesser summoners only practice a Magical or Faerie version of Summoning, normally learned through a Mystery Initiation, and avoid the deeper Goetic Arts. These summoners are limited by being unable to coerce spirits into obedience, but the temptation to descend into the Ars Goetia is always there, ready for them to snatch it. Greater summoners practice the full spectrum of the Ars Goetia, often in preference to the Solomonic Arts.
Character Creation: Lesser summoners are normally Companion-level, and take Summoning and the Solomonic Art of their mentor as Major Virtues. Full-fledged Summoners take Infernal Tradition instead, learn all four Goetic Arts, and may be played as Companions or Mythic Companions. Gifted Summoners are not unheard of (though it's common for them to have the False Gift); they may take Infernal Tradition or Summoning as their free Summoning Art. Mulhidun sorcerers should normally be Wealthy, since they can openly conjure up demons for personal gain.
Mechanical Notes: Summoning can't be used to coerce a spirit (though it can be used for highly-effective Scouring), but Binding or Commanding can, or the summoner can try to strike a pact.
Holy Sahirs
Summoning Art: Invocation+Adjuration or Kabbalah+Adjuration (this is a special exemption to the one-Art rule, due to the use of Difficult Arts, and it's still weaker for the XP than a conventional Accelerated Ability-based summoning art).
Warping: Divine Ascent, unless you want Fatra Bayna for some reason.
The Power of Adjuration allows the summoning of beings from other Realms through the limitless power of God. Developing the necessary internal strength to invoke this power is a demanding task, but there are a couple of traditions that particularly specialize in the calling of angels and spirits to gain their aid and service. These traditions are the Kabbalists, Jewish sorcerers who study divine mysteries, and the Mobeds, priests of the Mazdean faith and ancient princes of magic. Some of the Mazdean priests have joined the Suhhar and begun to practice Solomonic magic, hoping to rediscover some of the power that they have lost. There may also be a few practitioners of Kabbalah who have taken up the Solomonic Arts, though many rabbis would consider this sinful. (However, as many rabbis would consider Kabbalah itself to be in violation of mitzvot, particularly the use of astrological correspondences inherent in the practice, this is not as serious a proscription as it would seem.)
Other holy traditions exist that allow for the summoning of spirits, but these are much less common; solitary ascetics may develop the power of Invocation, for instance, but are unlikely to spend enough time in society to join the Suhhar. If it fits the needs of your troupe, there may be a small Christian or Muslim mystical sect that teaches these powers, or a Mythic Companion might have developed the secrets of the angels through their own research.
Character Creation: Holy Sahirs are, almost without exception, Mythic Companions; Gifted mobeds and kabbalists are highly unusual. Mobeds are likely to take True Faith and Divine Tradition. Take one's chosen Solomonic Arts, additional Mazdean magic and Outsider, and season to taste. Consider a decent stock of True Names and the Guardian Angel virtue.
Kabbalists with True Faith are unusual, but if you're making a mage-equivalent PC, go ahead and give him True Faith, Divine Tradition and Adjuration as well as a Solomonic Art or two. Such a character would learn Kabbalah, Merkavah and Ceremony (and possibly Gematria, though a sahir might have better uses for his time), marking him as an important leader in the yeshiva. Regular kabbalist-sahirs who have not attained such a state would be more likely to initiate Kabbalah and Adjuration, and Merkavah if that was part of his duty; if not, he should have been taught True Names by a senior Kabbalist. Initiates of Merkavah alone would make passable companions or supergrogs for such a yeshiva (if it were wealthy enough to support them), but would not qualify for Solomonic magic. Outsider is mandatory, of course.
Mechanical Notes: Invocation, aside from summoning spells, can be used either to command a spirit or to enforce a contract. A Guardian Angel will likely agree to be used as a vessel for magic; a default Guardian Angel probably has Might 15 and makes an excellent khadim.