  We derive exact analytic solutions for density and velocity fields to all
orders in Eulerian standard perturbation theory for $\Lambda$CDM cosmology. In
particular, we show that density and velocity field kernels can be written in a
separable form in time and momenta at each perturbative order. The kernel
solutions are built from an analytic basis of momentum operators and their
time-dependent coefficients, which solve a set of recursive differential
equations. We also provide an exact closed perturbative solution for such
coefficients, expanding around the (quasi-)EdS approximation. We find that the
perturbative solution rapidly converges towards the numerically obtained
solutions and its leading order result suffices for any practical requirements.
To illustrate our findings, we compute the exact two-loop dark matter density
and velocity power spectra in $\Lambda$CDM cosmology. We show that the
difference between the exact $\Lambda$CDM and the (quasi-)EdS approximated
result can reach the level of several percent (at redshift zero, for
wavenumbers $k<1h/$Mpc). This deviation can be partially mitigated by
exploiting the degeneracy with the EFT counterterms. As an additional benefit
of our algorithm for the solutions of time-dependent coefficients, the
computational complexity of power spectra loops in $\Lambda$CDM is comparable
to the EdS case. In performing the two-loop computation, we devise an explicit
method to implement the so-called IR cancellations, as well as the
cancellations arising as a consequence of mass and momentum conservation.
