  We present the integral decomposition for the fundamental solution of the
generalized Cattaneo equation with both time derivatives smeared through
convoluting them with some memory kernels. For power-law kernels $t^{-\alpha}$,
$\alpha\in(0,1]$ this equation becomes the time fractional one governed by the
Caputo derivatives which highest order is 2. To invert the solutions from the
Fourier-Laplace domain to the space-time domain we use analytic methods based
on the Efross theorem and find out that solutions looked for are represented by
integral decompositions which tangle the fundamental solution of the standard
Cattaneo equation with non-negative and normalizable functions being uniquely
dependent on the memory kernels. Furthermore, the use of methodology arising
from the theory of complete Bernstein functions allows us to assign such
constructed integral decompositions the interpretation of subordination. This
fact is preserved in two limit cases built into the generalized Cattaneo
equations, i.e., either the diffusion or the wave equations. We point out that
applying the Efross theorem enables us to go beyond the standard approach which
usually leads to the integral decompositions involving the Gaussian
distribution describing the Brownian motion. Our approach clarifies puzzling
situation which takes place for the power-law kernels $t^{-\alpha}$ for which
the subordination based on the Brownian motion does not work if
$\alpha\in(1/2,1]$.
