  Representations of polynomial covariance type commutation relations by linear
integral operators on $L_p$ over measures spaces are investigated. Necessary
and sufficient conditions for integral operators to satisfy polynomial
covariance type commutation relations are obtained in terms of their kernels.
For important classes of polynomial covariance commutation relations associated
to arbitrary monomials and to affine functions, these conditions on the kernels
are specified in terms of the coefficients of the monomials and affine
functions. By applying these conditions, examples of integral operators on
$L_p$ spaces, with separable kernels representing covariance commutation
relations associated to monomials, are constructed for the kernels involving
multi-parameter trigonometric functions, polynomials, and Laurent polynomials
on bounded intervals. Commutators of these operators are computed and exact
conditions for commutativity of these operators in terms of the parameters are
obtained.
