  We deduce a non-linear commutator higher-spin (HS) symmetry algebra which
encodes unitary irreducible representations of the AdS group -- subject to a
Young tableaux $Y(s_1,\ldots ,s_k)$ with $k\geq 2$ rows -- in a $d$-dimensional
anti-de-Sitter space. Auxiliary representations for a deformed non-linear HS
symmetry algebra in terms of a generalized Verma module, as applied to
additively convert a subsystem of second-class constraints in the HS symmetry
algebra into one with first-class constraints, are found explicitly in the case
of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg
algebra for the Verma module is constructed. The results generalize the method
of constructing auxiliary representations for the symplectic $sp(2k)$ algebra
used for mixed-symmetry HS fields in flat spaces \cite{BRbos}. Polynomial
deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied
as a by-product. A nilpotent BRST operator for a non-linear HS symmetry algebra
of the converted constraints for $Y(s_1, s_2)$ is found, with non-vanishing
terms (resolving the Jacobi identities) of third order in powers of ghost
coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation
for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced.
Following the results of \cite{BuchbinderReshetnyak, BRmasscub}, we develop a
BRST approach to constructing general off-shell local cubic interaction
vertices for irreducible massive higher-spin fields (being candidates for
massive particles in the Dark Matter problem). A new reducible gauge-invariant
Lagrangian formulation for an antisymmetric massive tensor field of spin
$(1,1)$ is obtained.
