  In this paper we show how the method of Zakharov transformations may be used
to analyze the stationary solutions of the Smoluchowski aggregation equation
for arbitrary homogeneous kernel. The resulting massdistributions are of
Kolmogorov type in the sense that they carry a constant flux of mass from small
masses to large. We derive a ``locality criterion'', expressed in terms of the
asymptotic properties of the kernel, that must be satisfied in order for the
Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads
to a gelation transition or not can be determined by computing the mass
capacity of the Kolmogorov spectrum. As an example, we compute the exact
stationary state for the family of
kernels,$K_\zeta(m_1,m_2)=(m_1m_2)^{\zeta/2}$ which includes both gelling and
non-gelling cases, reproducing the known solution in the case $\zeta=0$.
Surprisingly, the Kolmogorov constant is the same for all kernels in this
family.
