  None of the first-order modal logics between $\mathsf{K}$ and $\mathsf{S5}$
under the constant domain semantics enjoys Craig interpolation or projective
Beth definability, even in the language restricted to a single individual
variable. It follows that the existence of a Craig interpolant for a given
implication or of an explicit definition for a given predicate cannot be
directly reduced to validity as in classical first-order and many other logics.
Our concern here is the decidability and computational complexity of the
interpolant and definition existence problems. We first consider two decidable
fragments of first-order modal logic $\mathsf{S5}$: the one-variable fragment
$\mathsf{Q^1S5}$ and its extension $\mathsf{S5}_{\mathcal{ALC}^u}$ that
combines $\mathsf{S5}$ and the description logic$\mathcal{ALC}$ with the
universal role. We prove that interpolant and definition existence in
$\mathsf{Q^1S5}$ and $\mathsf{S5}_{\mathcal{ALC}^u}$ is decidable in
coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is
undecidable. These results transfer to the two-variable fragment
$\mathsf{FO^2}$ of classical first-order logic without equality. We also show
that interpolant and definition existence in the one-variable fragment
$\mathsf{Q^1K}$ of first-order modal logic $\mathsf{K}$ is non-elementary
decidable, while uniform interpolant existence is again undecidable.
