Extended Scalar Particle Solutions in Black String Spacetimes with Anisotropic Quintessence

We present novel solutions to the Klein-Gordon equation in a black string spacetime immersed in an anisotropic quintessence fluid and surrounded by a cloud of strings (BCK spacetime). The solutions are shown to depend on the quintessence state parameter $\alpha_{Q}$ and extend to a broader radial domain than previously reported. Explicit analytical results are provided for $\alpha_{Q} = 0, 1/2, 1$, thereby encompassing all relevant physical scenarios. These analytical radial solutions are derived using the confluent and biconfluent Heun equations, as well as Bessel equations in specific cases. Constraints on the Heun parameters that produce spectral restrictions are examined, resulting in the $\delta_{N}-$subclass of solutions, which includes the polynomial cases. Quintessence-induced \enquote{dark phases} are identified for several scenarios; in particular, the regime of $\alpha_{Q} = 1$ is emphasized for future comparison with alternative spacetime geometries. These findings contribute to the understanding of scalar particle dynamics and the influence of dark energy on quantum systems in curved spacetime backgrounds.