It is shown that the list of unusual mesons planned for a careful study in
photoproduction can be extended by the exotic states X±(1600) with IG(JPC)=2+(2++) which should be looked for in the ρ±ρ0 decay
channels in the reactions γN→ρ±ρ0N and γN→ρ±ρ0Δ. The full classification of the ρ±ρ0 states by their
quantum numbers is presented. A simple model for the spin structure of the γp→f2(1270)p, γp→a20(1320)p, and γN→X±(N,Δ) reaction amplitudes is formulated and the tentative estimates of the
corresponding cross sections at the incident photon energy Eγ≈6
GeV are obtained: σ(γp→f2(1270)p)≈0.12μb, σ(γp→a20(1320)p)≈0.25μb, σ(γN→X±N→ρ±ρ0N)≈0.018μb, and σ(γp→X−Δ++→ρ−ρ0Δ++)≈0.031μb. The problem of the X±
signal extraction from the natural background due to the other π±π0π+π− production channels is discussed. In particular the estimates are
presented for the γp→h1(1170)π+n, γp→ρ′+n→π+π0π+π−n, and γp→ωρ0p reaction cross sections.
Our main conclusion is that the search for the exotic X±(2+(2++))
states is quite feasible at JEFLAB facility. The expected yield of the γN→X±N→ρ±ρ0N events in a 30-day run at the 100% detection
efficiency approximates 2.8×106 events.Comment: 19 pages, revtex, 1 figure in postscipt, some comments and references
added, a few minor typos corrected, to be published in Phys. Rev.