  We systematically study the construction of mutually unbiased bases in
$\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$, such that all the bases are
unextendible maximally entangled ones. Necessary conditions of constructing a
pair of mutually unbiased unextendible maximally entangled bases in
$\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$ are derived. Explicit examples are
presented.
