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Let u: $$ {\mathbb{R}^2} \to {\mathbb{R}^2} $$ denote an entire solution of the homogeneous Euler-Lagrange equation associated to the energy used in the deformation theory of plasticity with logarithmic hardening. If |u(x)| is of slower growth than |x| as |x| → ∞, then u must be constant. Moreover, we show that u is affine if either $$ \mathop {{\sup }}\limits_{{\mathbb{R}^2}} |\nabla u|\, < \,\infty $$ or $$ \mathop {{{ \lim }\,{ \sup }}}\limits_{|x| \to \infty } |x{|^{ - 1}}|u(x)|\, < \,\infty $$. Bibliography: 13 titles.

On entire solutions of the equations for the displacement fields in the deformation theory of plasticity with logarithmic hardening