If (as your question seems to say) that you have three parameters to estimate ($\beta_0,\beta_1,a$) then this model is not identifiable.
\begin{eqnarray}
y&=&\beta_0+\beta_1(x_1-a)+u\\
&=&(\beta_0-\beta_1 a)+\beta_1 x_1+u\\
&=&\beta_0^*+\beta_1 x_1+u
\end{eqnarray}
Now consider that there's some best estimate of slope $\hat{\beta}_1$, and a corresponding best intercept $\hat{\beta}_0^*$ (this doesn't rely on how they're estimated - it doesn't matter).
Then in your formulation the intercept (given the slope) has two free parameters ($\beta_0$ and $a$), so that for a given $\hat{\beta}_0^*$ there's an infinite set of pairs ($\hat{\beta}_0$ and $\hat a$) which will give that best fitting value of intercept. So you simply can't tell which pair it might be -- they will all give identical fits.
As it stands, there's nothing to be done here.