  We study an optimization problem for a portfolio with a risk-free, a liquid,
and an illiquid risky asset. The illiquid risky asset is sold in an exogenous
random moment with a prescribed liquidation time distribution. The investor
prefers a negative or a positive exponential utility function. We prove that
both cases are connected by a one-to-one analytical substitution and are
identical from the economic, analytical, or Lie algebraic points of view.
  It is well known that the exponential utility function is connected with the
HARA utility function through a limiting procedure if the parameter of the HARA
utility function is going to infinity. We show that the optimization problem
with the exponential utility function is not connected to the HARA case by the
limiting procedure and we obtain essentially different results.
  For the main three dimensional PDE with the exponential utility function we
obtain the complete set of the nonequivalent Lie group invariant reductions to
two dimensional PDEs according to an optimal system of subalgebras of the
admitted Lie algebra. We prove that in just one case the invariant reduction is
consistent with the boundary condition. This reduction represents a significant
simplification of the original problem.
