  In this article, we formulate necessary and sufficient polynomial equations
for the existence of a symmetry plane or an order-two axial symmetry for a
totally symmetric tensor of order n $\ge$ 1. These conditions are effective and
of degree n (the tensor's order) in the components of the normal to the plane
(or the direction of the axial symmetry). These results are then extended to
obtain necessary and sufficient polynomial conditions for the existence of such
symmetries for an Elasticity tensor, a Piezo-electricity tensor or a
Piezo-magnetism pseudo-tensor.
