  The Navier-Stokes (NS) problem consists of finding a vector-function $v$ from
the Navier-Stokes equations. The solution $v$ to NS problem is defined in this
paper as the solution to an integral equation. The kernel $G$ of this equation
solves a linear problem which is obtained from the NS problem by dropping the
nonlinear term $(v \cdot \nabla)v$. The kernel $G$ is found in closed form.
Uniqueness of the solution to the integral equation is proved in a class of
solutions $v$ with finite norm $N_1(v)=\sup_{\xi\in \mathbb{R}^3, t\in [0,
T]}(1+|\xi|)(|v|+|\nabla v|)\le c (*)$, where $T>0$ and $C>0$ are arbitrary
large fixed constants. In the same class of solutions existence of the solution
is proved under some assumption. Estimate of the energy of the solution is
given.
