  The usual Helmholtz decomposition gives a decomposition of any vector valued
function into a sum of gradient of a scalar function and rotation of a vector
valued function under some mild condition. In this paper we show that the
vector valued function of the second term i.e. the divergence free part of this
decomposition can be further decomposed into a sum of a vector valued function
polarized in one component and the rotation of a vector valued function also
polarized in the same component. Hence the divergence free part only depends on
two scalar functions. Further we show the so called completeness of
representation associated to this decomposition for the stationary wave field
of a homogeneous, isotropic viscoelastic medium. That is by applying this
decomposition to this wave field, we can show that each of these three scalar
functions satisfies a Helmholtz equation. Our completeness of representation is
useful for solving boundary value problem in a cylindrical domain for several
partial differential equations of systems in mathematical physics such as
stationary isotropic homogeneous elastic/viscoelastic equations of system and
stationary isotropic homogeneous Maxwell equations of system. As an example, by
using this completeness of representation, we give the solution formula for
torsional deformation of a pendulum of cylindrical shaped homogeneous isotropic
viscoelastic medium.
