Possibilities to carry out low energy experiments (at E$^{\operatorname{lab}}$ $\sim$ 2--10 GeV) on the searches and studies of exotic hadrons and rare decays of mesons are considered: 1) search for baryonic states of $\mathrm{qqqs\bar s}$, $\mathrm{qqss\bar s}$, and $\mathrm{qqqg}$ types in direct channel; 2) search for exotic mesons in baryon exchange processes pN $\to$ (M)$_f$ + (pN;p$\pi$N); 3) study of exotic mesons in OPE hadron reactions and in photoproduction processes; 4) study of the rare electromagnetic decays of mesons; 5) study of the isotopic invariance violation.