  We construct two mutually unbiased bases by maximally entangled states
(MUMEB$s$) in $\mathbb{C}^{2}\otimes \mathbb{C}^{3}$. This is the first example
of MUMEB$s$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ when $d\nmid d'$, namely
$d'$ is not divisible by $d$. We show that they cannot be extended to four MUBs
in $\mathbb{C}^6$. We propose a recursive construction of mutually unbiased
bases formed by special entangled states with a fixed Schmidt number $k$
(MUSEB$k$s). It shows that
  $\min \{t_{1},t_{2}\}$ MUSEB$k_{1}k_{2}$s in $\mathbb{C}^{pd}\otimes
\mathbb{C}^{qd'}$ can be constructed from $t_{1}$ MUSEB$k_{1}$s in
$\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ and $t_{2}$ MUSEB$k_{2}$s in
$\mathbb{C}^{p}\otimes \mathbb{C}^{q}$ for any $d,d',p,q$. Further, we show
that three MUMEB$s$ exist in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ for any
$d,d'$ with $d\mid d'$, and two MUMEB$s$ exist in $\mathbb{C}^{d}\otimes
\mathbb{C}^{d'}$ for infinitely many $d,d'$ with $d\nmid d'$.
