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Purpose – The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold. Design/methodology/approach – The authors have used the tensorial approach to achieve the goal. Findings – A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g. Originality/value – The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.