  By definition, reciprocal matrices are tridiagonal $n$-by-$n$ matrices $A$
with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for
$i=1,\ldots,n-1$. For $n\leq 6$, we establish criteria under which the
numerical range generating curves (also called Kippenhahn curves) of such
matrices consist of elliptical components only. As a corollary, we also provide
a complete description of higher-rank numerical ranges when the criteria are
met.
