  This paper proposes an algorithm for deciding consistency of systems of
Boolean equations in several variables with co-efficients in the two element
Boolean algebra $B_{0}=\{0,1\}$ and find all satisfying assignments. The
algorithm is based on the application of a well known generalized Boole-Shannon
orthonormal (ON) expansion of Boolean functions. A necessary and sufficient
consistency condition for a special class of functions was developed in
\cite{sule} using such an expansion. Paper \cite{sule} develops a condition for
consistency of the equation $f(X)=0$ for the special classes of Boolean
functions 1) $f$ in $B(\Phi(X))$ for an ON set $\Phi$ of Boolean functions in
$X$ over a general Boolean algebra $B$ and 2) $f$ in $B(X_{2})(\Phi(X_{1}))$.
The present paper addresses the problem of obtaining the consistency conditions
for arbitrary Boolean functions in $B_{0}(X)$. Next, the consistency for a
single equation is shown equivalent to another system of Boolean equations
which involves the ON functions and characterizes all solutions. This result is
then extended for Boolean systems in several variables over the algebra
$B_{0}=\{0,1\}$ which does not convert the system into a single equation. This
condition leads to the algorithm for computing all solutions of the Boolean
system without using analogous resolution and determine satisfiability. For
special systems defined by CNF formulas this algorithm results into an
extension of the DPLL algorithm in which the \emph{splitting rule} is
generalized to several variables in terms of ON terms in the sense that
splitting of CNF set in a single variable $x$ is equivalent to ON terms $x,x'$.
