  It is shown that the list of unusual mesons planned for a careful study in
photoproduction can be extended by the exotic states $X^\pm(1600)$ with $I^G(J
^{PC})=2^+(2^{++})$ which should be looked for in the $\rho^\pm\rho^0$ decay
channels in the reactions $\gamma N\to\rho^\pm\rho^0N$ and $\gamma N\to\rho^\pm
\rho^0\Delta$. The full classification of the $\rho^\pm\rho^0$ states by their
quantum numbers is presented. A simple model for the spin structure of the $
\gamma p\to f_2(1270)p$, $\gamma p\to a^0_2(1320)p$, and $\gamma N\to X^\pm (N,
\Delta)$ reaction amplitudes is formulated and the tentative estimates of the
corresponding cross sections at the incident photon energy $E_\gamma\approx 6$
GeV are obtained: $\sigma(\gamma p\to f_2(1270)p)\approx0.12$ $\mu$b, $
\sigma(\gamma p\to a^0_2(1320)p)\approx0.25$ $\mu$b, $\sigma(\gamma N\to X^\pm
N\to\rho^\pm\rho^0N)\approx0.018$ $\mu$b, and $\sigma(\gamma p\to X^-\Delta^{++
}\to\rho^-\rho^0\Delta^{++})\approx0.031$ $\mu$b. The problem of the $X^\pm$
signal extraction from the natural background due to the other $\pi^\pm\pi^0
\pi^+\pi^-$ production channels is discussed. In particular the estimates are
presented for the $\gamma p\to h_1(1170)\pi^+n$, $\gamma p\to\rho'^{+}n\to
\pi^+\pi^0\pi^+\pi^-n$, and $\gamma p\to\omega\rho^0p$ reaction cross sections.
Our main conclusion is that the search for the exotic $X^\pm(2^+(2^{++}))$
states is quite feasible at JEFLAB facility. The expected yield of the $\gamma
N\to X^\pm N\to\rho^\pm\rho^0N$ events in a 30-day run at the 100% detection
efficiency approximates $2.8\times10^6$ events.
