  We report a combined experimental and theoretical study of the dependence of
the superconductivity of the unconventional superconductor Sr$_2$RuO$_4$ on
anisotropic strain. Novel piezoelectric apparatus is used to apply uniaxial
pressures of up to $\sim$1~GPa along a $\langle 100 \rangle$ direction
($a$-axis) of the crystal lattice. $T_c$ increases from 1.5~K in unstrained
material to 3.4~K at compression by $\approx$0.6\%, then falls steeply. The
$c$-axis upper critical field for the strained $T_c$ = 3.4~K material is a
factor of twenty larger than that of the unstrained crystal, whereas the
in-plane ($a$-axis) critical field increases by only a factor of three.
First-principles electronic structure calculations give evidence that the
observed maximum $T_c$ occurs at or near a Lifshitz transition when the Fermi
level passes through a van Hove singularity. Finally, we perform order
parameter analyses using three-band renormalization group calculations. These,
combined with the unexpectedly low in-plane critical field, open the
possibility that the highly strained $T_c$=3.4~K Sr$_2$RuO$_4$ has an even-
rather than an odd-parity order parameter. Potential implications such as a
transition at nonzero strain between odd- and even-parity order parameters are
discussed.

Superconductivity