  We consider the elastic theory for Ising transitions in an isotropic elastic
medium in the zero thermal expansion (ZTE) limit. We use this theory to study
the nature of the fluctuations in the system near the second phase transitions
at $T_c$ in the ZTE limit given by $dT_c/dV=0$, where $V$ is the system volume,
and explore anomalous elasticity. Allowing for the local strain to couple {\em
asymmetrically} with the states of the order parameter, we uncover the dramatic
effects of these couplings on the fluctuations of the local displacements near
$T_c$, and also on the nature of the phase transition itself. Near second order
phase transitions and with weak asymmetry in the order parameter - strain
couplings, the variance of the displacement fluctuations in two dimensions
scale with the system size $L$ in a universal fashion as $[\ln (L/a_0)]^{2/3}$;
$a_0$ is a small-scale cutoff. Likewise, the correlation functions of the
difference of the local displacements at two different points separated by $r$
scale as $[\ln (r/a_0)]^{2/3}$ for large $r$. For stronger asymmetry, this
variance diverges as $L$ exceeds beyond a (nonuniversal) size, determined by
the model parameters, signaling a transition to a phase with only short range
order or the loss of the positional order of the elastic medium. At dimensions
higher than two, for sufficiently weak selectivity, the variance of the
displacement fluctuations is $L$-independent corresponding to long range order.
However, if the selectivity parameters rise beyond a dimension-dependent
threshold values, again the positional order is lost with a concomitant
transition to a phase with short range order. Large values of the order
parameter - strain couplings can turn the phase transition into a first order
as well.
