Exact solvability of brane equations is studied, and a new U(1) x U(I) x ... x U(1) invariant anzats for the solution of p-brane equations in D = (2p + 1)-dimensional Minkowski space is proposed. The reduction of the p-brane Hamiltonian to the Hamiltonian of p-dimensional relativistic anharmonic oscillator with the monomial potential of the degree equal to 2p is revealed. For the case of degenerate p-torus with equal radii it is shown that the p-brane equations are integrable and their solutions are expressed in terms of elliptic (p = 2) or hyperelliptic (p > 2) functions. The solution describes contracting p-brane with the contraction time depending on p and the brane energy density. The toroidal brane elasticity is found to break down linear Hooke law as it takes place for the anharmonic elasticity of smectic liquid crystals.