  The concept of bidemocratic pair for a Banach space was introduced in
\cite{KS:18}. We construct a family of orthonormal systems $\mathfrak{F}_{l},$
$l\in (0,\infty)$ of functions defined on $[-1,1]$ such that the pair
$(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is bidemocratic for $L^{p}[-1,1]$ and for
$L^{p'}[-1,1]$ if $l\in (0, \frac{p}{2(p-2)}]$, where $p>2$ and $p'=
\frac{p}{p-1}$. The system $\mathfrak{F}_{l}$ is not democratic for
$L^{p'}[-1,1]$ when $l\in (\frac{p}{2(p-2)}, \frac{p}{p-2}). $ When $l>
\frac{p}{2(p-2)}$ the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is not
bidemocratic neither for $L^{p}[-1,1]$ nor for $L^{p'}[-1,1]$.
