  In this paper we will show the existence of a face $2$-colourable biembedding
of the complete graph onto an orientable surface where each face is a cycle of
a fixed length $k$, for infinitely many values of $k$. In particular, under
certain conditions, we show that there exists at least $(n-2)[(p-2)!]^2/(e^2
kn)$ non-isomorphic face $2$-colourable biembeddings of $K_{2nk+1}$ in which
all faces are cycles of length $k=4p+3$. These conditions are: $n\equiv 1\mod
4$, $k\equiv 3\mod 4$ and either $n$ is prime or $n\gg k$ and $n\equiv 0\mod 3$
implies $p\equiv 1\mod 3$. To achieve this result we begin by verifying the
existence of $(n-2)[(p-2)!/e]^2$ non-equivalent Heffter arrays, $H(n;k)$, which
satisfy the conditions: (1) for each row and each column the sequential partial
sums determined by the natural ordering must be distinct modulo $2nk+1$; (2)
the composition of the natural orderings of the rows and columns is equivalent
to a single cycle permutation on the entries in the array. The existence of
Heffter arrays $H(n;k)$ that satisfy condition (1) was established earlier in
\cite{BCDY} and in this current paper we vary this construction and show that
there are at least $(n-2)[(p-2)!/e]^2$ such non-equivalent $H(n;k)$ that
satisfy condition (1) and then show that each of these Heffter arrays also
satisfy condition (2) under certain conditions.
