To solve boundary value problems in micropolar elasticity theory, the work formulates a Lagrange variational principle in generalized kinematic fields applicable to centrally symmetric materials with arbitrary anisotropy. Using the Ritz method, the boundary value problem is reduced to a tensor-block system of linear algebraic equations. This is achieved by expanding the sought kinematic vector fields of displacements and microrotations into series of piecewise-polynomial basis functions from Lagrange and Serendipity families. To improve approximation by Lagrange polynomials, including for nearly incompressible media, the generalized method of reduced and selective integration is employed. The developed model is validated using the torsion problem of an isotropic circular cylinder within both classical and micropolar elasticity theories, demonstrating scale effects and subsequent comparison with experimental results. When setting integral boundary conditions on the cylinder’s end surface, the distribution of tangential and moment stresses obtained from analytical solution was used.