  We present the general theory of Ising transitions in isotropic elastic media
with vanishing thermal expansion. By constructing a minimal model with
appropriate spin-lattice couplings, we show that in two dimensions near a
continuous transition the elasticity is anomalous in unusual ways: the system
either significantly stiffens with a hitherto unknown unique positional order
logarithmically stronger than quasi-long range order, or, as the
inversion-asymmetry of the order parameter in its coupling with strain
increases, it destabilizes when system size $L$ exceeds a finite threshold. At
three dimensions, stronger inversion-asymmetric couplings induce instability to
the long-range positional order for all $L$. Sufficiently strong order
parameter-displacement couplings can also turn the phase transition first order
at all dimensions, concomitant with finite jumps in the elastic modulii across
the transition. Our theory establishes a {\em one-to-one correspondence}
between the order of the phase transitions and anomalous elasticity near the
transitions.
