  There exist two conjectures for constraint satisfaction problems (CSPs) of
reducts of finitely bounded homogeneous structures: the first one states that
tractability of the CSP of such a structure is, when the structure is a
model-complete core, equivalent to its polymorphism clone satisfying a certain
non-trivial linear identity modulo outer embeddings. The second conjecture,
challenging the approach via model-complete cores by reflections, states that
tractability is equivalent to the linear identities (without outer embeddings)
satisfied by its polymorphisms clone, together with the natural uniformity on
it, being non-trivial.
  We prove that the identities satisfied in the polymorphism clone of a
structure allow for conclusions about the orbit growth of its automorphism
group, and apply this to show that the two conjectures are equivalent. We
contrast this with a counterexample showing that $\omega$-categoricity alone is
insufficient to imply the equivalence of the two conditions above in a
model-complete core.
  Taking a different approach, we then show how the Ramsey property of a
homogeneous structure can be utilized for obtaining a similar equivalence under
different conditions.
  We then prove that any polymorphism of sufficiently large arity which is
totally symmetric modulo outer embeddings of a finitely bounded structure can
be turned into a non-trivial system of linear identities, and obtain
non-trivial linear identities for all tractable cases of reducts of the
rational order, the random graph, and the random poset.
  Finally, we provide a new and short proof, in the language of monoids, of the
theorem stating that every $\omega$-categorical structure is homomorphically
equivalent to a model-complete core.
