  The symmetries of flow structures are often prescribed by their mechanical
instability and geometry. Here, as an example, we present the homotopy of the
rotating 3-fold spiral state that is robust in a spherical Couette flow towards
the hybrid system with a thermal stratification effect. It has not yet been
confirmed that the rotating wave state smoothly connects to the thermal
stratification system. Through continuation, the most dangerous mode at a
purely spherical Couette flow of $m=4$ modes of spherical harmonics is replaced
by $l=4$ and $m=3$ in a purely thermal convective system. For the state
obtained at the limit under only the thermal effect, the residual quantities of
both the torque to the outer sphere and meridional circulation are discussed in
detail.
