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The aim of the article to clarify the status of Shapiro plane wave solutionsof the Schr\"odinger's equation in the frames of the well-known general methodof separation of variables. To solve this task, we use the well-knowncylindrical coordinates in Riemann and Lobachevsky spaces, naturally relatedwith Euler angle-parameters. Conclusion may be drawn: the general method ofseparation of variables embraces the all plane wave solutions; the plane wavesin Lobachevsky and Riemann space consist of a small part of the whole set ofbasis wave functions of Schr\"odinger equation.  In space of constant positive curvature $S_{3}$, a complex analog ofhorospherical coordinates of Lobachevsky space $H_{3}$ is introduced. Toparameterize real space $S_{3}$, two complex coordinates $(r,z)$ must obeyadditional restriction in the form of the equation $r^{2} = e^{z-z^{*}} -e^{2z} $. The metrical tensor of space $S_{3}$ is expressed in terms of $(r,z)$with additional constraint, or through pairs of conjugate variables $(r,r^{*})$or $(z,z^{*})$; correspondingly exist three different representations forSchr\"{o}dinger Hamiltonian. Shapiro plane waves are determined and explored assolutions of Schr\"odinger equation in complex horosperical coordinates of$S_{3}$. In particular, two oppositely directed plane waves may be presented asexponentials in conjugated coordinates. $\Psi_{-}= e^{-\alpha z}$ and$\Psi_{+}= e^{-\alpha z^{*}}$. Solutions constructed are single-valued, finite,and continuous functions in spherical space and correspond to discrete energylevels.

Shapiro's plane waves in spaces of constant curvature and separation of variables in real and complex coordinates