  Management of a portfolio that includes an illiquid asset is an important
problem of modern mathematical finance. One of the ways to model illiquidity
among others is to build an optimization problem and assume that one of the
assets in a portfolio can not be sold until a certain finite, infinite or
random moment of time. This approach arises a certain amount of models that are
actively studied at the moment.
  Working in the Merton's optimal consumption framework with continuous time we
consider an optimization problem for a portfolio with an illiquid, a risky and
a risk-free asset. Our goal in this paper is to carry out a complete Lie group
analysis of PDEs describing value function and investment and consumption
strategies for a portfolio with an illiquid asset that is sold in an exogenous
random moment of time with a prescribed liquidation time distribution. The
problem of such type leads to three dimensional nonlinear
Hamilton-Jacobi-Bellman (HJB) equations. Such equations are not only tedious
for analytical methods but are also quite challenging form a numeric point of
view. To reduce the three-dimensional problem to a two-dimensional one or even
to an ODE one usually uses some substitutions, yet the methods used to find
such substitutions are rarely discussed by the authors.
  We find the admitted Lie algebra for a broad class of liquidation time
distributions in cases of HARA and log utility functions and formulate
corresponding theorems for all these cases. We use found Lie algebras to obtain
reductions of the studied equations. Several of similar substitutions were used
in other papers before whereas others are new to our knowledge. This method
gives us the possibility to provide a complete set of non-equivalent
substitutions and reduced equations.
