In this paper, we consider the CPE conjecture in the frame-work of \({K}\)-contact manifold and \({(\kappa, \mu)}\)-contact manifold. First, we prove that a complete \({K}\)-contact metric satisfying the CPE is Einstein and is isometric to a unit sphere \({S^{2n+1}}\). Next, we prove that if a non-Sasakian \({ (\kappa, \mu) }\)-contact metric satisfies the CPE, then \({ M^{3} }\) is flat and for \({ n > 1 }\), \({ M^{2n+1} }\) is locally isometric to \({ E^{n+1} \times S^{n}(4)}\).

The prime objective of the approach is to give geometric classifications of k -almost Ricci solitons associated with paracontact manifolds. Let M 2 n + 1 ( φ , ξ , η , g ) be a paracontact metric manifold, and if a K -paracontact metric g represents a k -almost Ricci soliton ( g , V , k , λ ) and the potential vector field V is Jacobi field along the Reeb vector field ξ , then either k = λ − 2 n , or g is a k -Ricci soliton. Next, we consider K -paracontact manifold as a k -almost Ricci soliton with the potential vector field V is infinitesimal paracontact transformation or collinear with ξ . We have proved that if a paracontact metric as a k -almost Ricci soliton associated with the non-zero potential vector field V is collinear with ξ and the Ricci operator Q commutes with paracontact structure φ , then it is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) . Finally, we have deduced that a para-Sasakian manifold admitting a gradient k -almost Ricci soliton is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) .