Turkish troops are not authorized to be on the territory of the Syrian Arab Republic from the point of view of international law. Official Damascus has sent a formal request for military aid only to Moscow. However, until recently, the Assad government has reacted to the actions of Turkish troops on Syrian territory very discreetly. Following the normalization of relations between Moscow and Turkey, none of these countries is interested in a new spiral of conflicts, especially Damascus.
The recent incident between Syria and Turkey is likely a provocation, just like the destruction of the Russian SU-24 at the end of 2015. Turkish authorities have established that pro-Western organizations stood for the murder of the Russian pilot. The current provocation was possibly organized in order to destabilize and plunge the region into even greater chaos.
If the Turkish and Syrian sides will not be able to resolve the situation in a realist way, then the conflict could possibly go to a new level. The only force interested in such a development is the neocons and globalists who still control the main centers of power in the West, primarily in the United States.