О представлении системы полулинейных и полуквадратичных соотношений

We study the $\ast$-algebra which generated by two selfadjoint elements $a$, $b$ satisfying the algebraic relations:
\[
\sum_{i=1}^mf_i(a)bg_i(a) = h(a),\quad\sum_{j=1}^mp_j(a)br_j(a)bq_j(a) = v(a),
\]
where $f_i$, $g_i$, $h$, $i = \overline{1, m}$, $p_j$, $r_j$, $q_j$, $v$, $j = \overline{1, l}$ are polynomials on $\mathbb{R}$, $m$, $l \in \mathbb{N}$. We investigate properties of polynomials $f_i$, $g_i$, $h$, $i = \overline{1, m}$, $p_j$, $r_j$, $q_j$, $v$, $j = \overline{1, l}$ for which this $\ast$-algebra is $\ast$-tame. The results are illustrated by examples.