  The Exact Satisfiability problem asks if we can find a satisfying assignment
to each clause such that exactly one literal in each clause is assigned $1$,
while the rest are all assigned $0$. We can generalise this problem further by
defining that a $C^j$ clause is solved iff exactly $j$ of the literals in the
clause are $1$ and all others are $0$. We now introduce the family of
Generalised Exact Satisfiability problems called G$i$XSAT as the problem to
check whether a given instance consisting of $C^j$ clauses with $j \in
\{0,1,\ldots,i\}$ for each clause has a satisfying assignment. In this paper,
we present faster exact polynomial space algorithms, using a nonstandard
measure, to solve G$i$XSAT, for $i\in \{2,3,4\}$, in $O(1.3674^n)$ time,
$O(1.5687^n)$ time and $O(1.6545^n)$ time, respectively, using polynomial
space, where $n$ is the number of variables. This improves the current state of
the art for polynomial space algorithms from $O(1.4203^n)$ time for G$2$XSAT by
Zhou, Jiang and Yin and from $O(1.6202^n)$ time for G$3$XSAT by Dahll\"of and
from $O(1.6844^n)$ time for G$4$XSAT which was by Dahll\"of as well. In
addition, we present faster exact algorithms solving G$2$XSAT, G$3$XSAT and
G$4$XSAT in $O(1.3188^n)$ time, $O(1.3407^n)$ time and $O(1.3536^n)$ time
respectively at the expense of using exponential space.
