  We study the computational complexity of deciding whether a given set of term
equalities and inequalities has a solution in an $\omega$-categorical algebra
$\mathfrak{A}$. There are $\omega$-categorical groups where this problem is
undecidable. We show that if $\mathfrak{A}$ is an $\omega$-categorical
semilattice or an abelian group, then the problem is in P or NP-hard. The hard
cases are precisely those where Pol$(\mathfrak{A},\neq)$ has a uniformly
continuous minor-preserving map to the clone of projections on a two-element
set. The results provide information about algebras $\mathfrak{A}$ such that
Pol$(\mathfrak{A},\neq)$ does not satisfy this condition, and they are of
independent interest in universal algebra. In our proofs we rely on the
Barto-Pinsker theorem about the existence of pseudo-Siggers polymorphisms. To
the best of our knowledge, this is the first time that the pseudo-Siggers
identity has been used to prove a complexity dichotomy.
