  We investigate nonanalyticities (e.g., zeros and poles) of refractive index
$n(\omega)$ and group index $n_g(\omega)$ in different optical setups. We first
demonstrate that: while a Lorentzian dielectric has no nonanalyticity in the
upper half of the complex frequency plane (CFP), its group index -- which
governs the pulse-center propagation -- violates the Kramers-Kronig relations
(KKRs). Thus, we classify the nonanalyticities as in the (a) first-order
(refractive index or reflection) and (b) second-order (group index or group
delay). The latter contains the derivative of the former. Then, we study a
possible connection between the negative superluminal velocities and the
presence of nonanalyticities in the upper half of the CFP. We show that
presence of nonanalyticities in the upper half of the CFP for (a) the
first-order response and (b) the second-order response are accompanied by the
appearance of negative (a) phase velocity and (b) group velocity, respectively.
We also distinguish between two kinds of superluminosity, $v>c$ and $v<0$,
where we show that the second one ($v<0$) appears with the violation of KKRs.
