  To investigate $pn$ pair correlations in $N=Z=\textrm{odd}$ nuclei, we
develop a new framework based on the generator coordinate method of the
$\beta\gamma$ constraint antisymmetrized molecular dynamics. In the framework,
the isospin projection is performed before the energy variation to obtain the
wave function optimized for each isospin. We apply the method to ${}^{10}
\textrm{B}$ and show that it works well to describe coexistence of $T=0$ and
$T=1$ states in low-energy spectra. Structures of low-lying states and $pn$
correlations are investigated. Strong $M1$($0^+_1\rightarrow 1^+_1$) and
$E2$($1^+_1\rightarrow 1^+_2$) transitions are understood by the spin
excitation of the $pn$ pair and the rotation of a deformed core, respectively.
