Thermosolutal instability of Veronis (1965) type in Rivlin-Ericksen viscoelastic fluid in the presence of uniform vertical rotation in a porous medium is considered. The paper established the condition for characterizing the oscillatory motions which may be neutral or unstable, for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. It is established that all non-decaying slow motions starting from rest, in a Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension and finite vertical depth in a porous medium, are necessarily non-oscillatory, in the regime ,__ampersandsignnbsp;where s R is the Thermosolutal Rayliegh number, A T is the Taylor number, 2 p is the magnetic Prandtl number, 3 p is the hermosolutal Prandtl number, l P is the medium permeability, ? is the porosity and F is the viscoelasticity parameter. The result is important since it hold for all wave numbers and for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. A similar characterization theorem is also proved for Stern (1960) type of configuration.