  The non-additive kinetic potential $v^{\text{NAD}}$ is a key quantity in
density-functional theory (DFT) embedding methods, such as frozen density
embedding theory and partition DFT. $v^{\text{NAD}}$ is a bi-functional of
electron densities $\rho_{\rm B}$ and $\rho_{\rm tot} = \rho_{\rm A} +
\rho_{\rm B}$. It can be evaluated using approximate kinetic-energy
functionals, but accurate approximations are challenging. The behavior of
$v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and
singularities were seen in some approximate calculations. In this article, the
existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for
various choices of $\rho_{\rm B}$ and $\rho_{\rm tot}$, using the nuclear cusp
conditions for the density and Kohn-Sham potential. It is shown that no
singularities arise from smoothly partitioned ground-state Kohn-Sham densities.
We confirm this result by numerical calculations on diatomic test systems HeHe,
HeLi$^+$, and H$_2$, using analytical inversion to obtain a numerically exact
$v^{\rm NAD}$ for the local density approximation. We examine features of
$v^{\rm NAD}$ which can be used for development and testing of approximations
to $v^{\rm NAD}[\rho_{\rm B},\rho_{\rm tot}]$ and kinetic-energy functionals.
