Spherical orders, properties and countable spectra of their theories

We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and decidable. The values for spectra of countable models of unary expansions of $n$-spherical theories are described. The Vaught conjecture is confirmed for countable constant expansions of dense $n$-spherical theories.