  We consider a simple model for QCD dynamics in which DGLAP
integro-differential equation may be solved analytically. This is a gauge model
which possesses dominant evolution of gauge boson (gluon) distribution and in
which the gauge coupling does not run. This may be ${\cal N} =4$ supersymmetric
gauge theory with softly broken supersymmetry, other finite supersymmetric
gauge theory with lower level of supersymmetry, or topological Chern-Simons
field theories. We maintain only one term in the splitting function of
unintegrated gluon distribution and solve DGLAP analytically for this
simplified splitting function. The solution is found by use of the Cauchy
integral formula. The solution restricts form of the unintegrated gluon
distribution as function of momentum transfer and of Bjorken $x$. Then we
consider an almost realistic splitting function of unintegrated gluon
distribution as an input to DGLAP equation and solve it by the same method
which we have developed to solve DGLAP equation for the toy-model. We study a
result obtained for the realistic gluon distribution and find a singular
Bessel-like behaviour in the vicinity of the point $x=0$ and a smooth behaviour
in the vicinity of the point
