  The paper conducts a second-order variational analysis for an important class
of nonpolyhedral conic programs generated by the so-called
second-order/Lorentz/ice-cream cone $Q$. From one hand, we prove that the
indicator function of $Q$ is always twice epi-differentiable and apply this
result to characterizing the uniqueness of Lagrange multipliers at stationary
points together with an error bound estimate in the general second-order cone
setting involving ${\cal C}^2$-smooth data. On the other hand, we precisely
calculate the graphical derivative of the normal cone mapping to $Q$ under the
weakest metric subregularity constraint qualification and then give an
application of the latter result to a complete characterization of isolated
calmness for perturbed variational systems associated with second-order cone
programs. The obtained results seem to be the first in the literature in these
directions for nonpolyhedral problems without imposing any nondegeneracy
assumptions.
