Two-stage Runge-Kutta family with an implicit second stage ($R(z)=\frac{1+(1-a)z+(1/2-a)z^2}{1-az}$)

For stages $k_1=f(y)$ and $k_2=f(y+(1-a)h{k}_1+ah{k}_2)$ with update $y+h(k_1+k_2)/2$, the method has order two for every fixed real $a$. The third-order coefficient of $f^{\prime \prime }{f}^2$ is always $1/4$, not $1/6$, so no parameter gives third order on general nonlinear equations. The displayed stability function is bounded throughout the left half-plane only when $a=1/2$, when it is the A-stable trapezoidal function $(1+z/2)/(1-z/2)$. It is not L-stable.