We impose single-channel unitarity on a five-term Veneziano $\pi -\pi$ amplitude using the $K$-matrix approach. The violation of crossing symmetry as measured by the $s$- and $p$-wave crossing-symmetry sum rules is found to be negligible for wide ranges in the satellite coefficients. We study the conditions imposed by the non-negativity of the widths of the $\rho$, $f$, and $g$ mesons as well as all their daughters. Although these conditions still permit a continuous infinity of solutions, a limiting case is of some interest where the widths of the ${\epsilon }^{\prime }$, ${\epsilon }^{\prime \prime }$, and $f^{\prime }$ are made to vanish. For this special model we calculate widths, scattering lengths, and the phase shifts $\delta _0^{}{}_{}{}^0$, $\delta _0^{}{}_{}{}^2$, and $\delta _1^{}{}_{}{}^1$. Since the daughters are no longer degenerate with the parents, masses are also calculated.

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