  Generalized Pauli's theorem, proved by D. S. Shirokov for two sets of
anticommuting elements of a real or complexified Clifford algebra of dimension
$2^n$, is extended to the case, when both sets of elements depend smoothly on
points of Euclidian space of dimension $r$. We prove that in the case of even
$n$ there exists a smooth function such that two sets of Clifford algebra
elements are connected by a similarity transformation. All cases of connection
between two sets are considered in the case of odd $n$. Using the equation for
the spin connection of general form, it is shown that the problem of the local
Pauli's theorem is equivalent to the problem of existence of a solution of some
special system of partial differential equations. The special cases $n=2$,
$r\geq 1$ and $n\geq 2$, $r=1$ with more simpler solution of the problem are
considered in detail.
