  In this paper the Levy-Leblond procedure for linearizing the Schr\"odinger
equation to obtain the Pauli equation for one particle is generalized to obtain
an $N$-particle equation with spin. This is achieved by using the more
universal matrix factorization, $G\tilde{G} = |G| I = (-K)^l I$. Here the
square matrix $G$ is linear in the total energy E and all momenta, $\tilde G$
is the matrix adjoint of $G$, $I$ is the identity matrix, $|G|$ is the
determinant of $G$, $l$ is a positive integer and $K=H-E$ is Lanczos' extended
Hamiltonian where $H$ is the classical Hamiltonian of the electro-mechanical
system. $K$ is identically zero for all such systems, so that matrix $G$ is
singular. As a consequence there always exists a vector function
$\underline\theta$ with the property $G\underline\theta=0$. This factorization
to obtain the matrix $G$ and vector function $\underline\theta$ is illustrated
first for a one-dimensional particle in a simple potential well. This same
technique, when applied to the classical nonrelativistic Hamiltonian for $N$
interacting particles in an electromagnetic field, is shown to yield for N=1
the Pauli wave equation with spin and its generalization to $N$ particles.
Finally this nonrelativistic generalization of the Pauli equation is used to
treat the simple Zeeman effect of a hydrogen-like atom as a two-particle
problem with spin.
