An algorithm to decide if an orientable atoroidal 3-manifold contains closed incompressible acylindrical surfaces, and construct closed incompressible acylindrical surfaces is given.Mainly, the normal surface theory is used.To assure that the algorithm stops after finite steps, we show that each acylindrical surface is isotopic to some ``edge surface'' which is constructible.
"An Algorithm for Acylindrical Surfaces in 3-manifolds." Tokyo J. Math. 24 (2) 395 - 405, December 2001. https://doi.org/10.3836/tjm/1255958183