Adding productivity shocks to the growth model turns a theory of accumulation into a theory of business cycles. Almost nothing in that literature has a closed form solution.
The Brock-Mirman case is the exception. Logarithmic utility, Cobb-Douglas production, and complete depreciation within the period together yield an exact policy rule, which is why we study this model closely. It is the benchmark against which numerical solutions to harder problems can be checked (a solver that cannot reproduce Brock-Mirman to machine precision has a bug, not an approximation error). The Prescott real business cycle model gives up that luxury in exchange for realism, and asks how much of observed output volatility technology shocks alone can account for.