I very strongly, sincerely doubt that there will be a cold war in the Middle East. I don't think Syria has the brains, funds, or patience to carry on a cold war with Israel. Most of the Islamic countries have periodically started hot wars with Israel, but I sincerely doubt that a cold one will happen.
If a cold war does develop, the world can be certain that Putin is holding the puppet strings there just as he seems to be with Medavev in Russia. Even then, it's more likely to be just another shooting, proxy war between the US and Russia, with Israel and Syria as the on the ground players, rather than the government and rebel forces in, say, El Salvador, in the '80s.
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