  In this paper we present the details of our previous work on exact solutions
in second-order conformal hydrodynamics together with a number of new solutions
found by mapping Minkowski space onto various curved spacetimes such as anti-de
Sitter space and hyperbolic space. We analytically show how the solutions of
ideal hydrodynamics are modified by the second-order effects including
vorticity. We also find novel boost-invariant exact solutions which exist only
in second-order hydrodynamics and have an unusual dependence on the proper
time.
