On representation varieties of one class of HNN extensions

We investigate representation varieties $R_n(G(p,q))$ of the groups with the following presentation:
$$ G(p,q) = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g,y_1,\ldots,y_g,t\mid a_1^{m_1}=\ldots=a_s^{m_s}=1,\ tU^pt^{-1}=U^q \rangle, $$
where $p$ and $q$ are such integers that $p>|q|\geq1$, $m_i\ge 2$ for $i=1,\ldots,s$, $g\ge 2$,$U=[x_1,y_1]\ldots [x_g,y_g]W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ and $W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ is a reduced word in the free product of cyclic groups $H=\langle a_1\mid a_1^{m_1}\rangle\ast\ldots\ast\langle a_s\mid a_s^{m_s}\rangle\ast\langle b_1\rangle\ast\ldots\ast \langle b_k\rangle$. Irreducible components of $R_n(G(p,q))$ are found, their dimensions are calculated and every irreducible component of $R_n(G(p,q))$ is proved to be a rational variety.