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Linear elasticity has long been considered a well-established research area using conservative field theory. However, the discovery of odd-elasticity challenges the essential energy conservation assumption, which together with gyroscopic ingredients compromise the fundamental theory of elasticity, but to the same effect, enable new directions in active elastodynamics. Here, we consider two-dimensional continuum mechanics in a more general framework containing active constituents from both gyroscopic and odd-elastic effects, which gives rise to non-reciprocal and non-Hermitian elastic waves in a highly unconventional guise. We discuss how these unusual media can extract energy from odd-elastic engine cycles comprising remarkable features of stability transitions, in which the energy exchange process reverses. Beyond bulk waves, akin to the unidirectional characteristics of a 2D quantum-Hall insulator, we demonstrate the existence of non-Hermitian Rayleigh surface waves which, in contrast to the classical ones in passive solids, display one-way and interference-free transport characteristics, which even remain resilient in finite sharp or curved geometries. The findings reported here may provide new possibilities to manipulate elastic waves in unusual ways.