Preheat oven to 180C/350F. Grease and line the bottom of a 23cm/11inch round pan.
With a stand mixer or hand held mixer whisk together eggs, sugar, olive oil and vanilla until thick and pale, about 5 minutes.
Slowly incorporate the lime juice and zest.
With the mixer on low alternately add flour and milk in three batches. Mix until on just combined. Don't overbeat.
Pour the cake batter into the prepared pan and evenly sprinkle blueberries over the surface.
Bake for 1 hour.
10 minutes before the cake is ready prepare the syrup.
Combine all ingredients for the syrup in a small saucepan. Bring to the boil then simmer for about 6-8 minutes or until reduced and syrupy.
Remove cake from the oven and slowly pour hot syrup over hot cake. You may need to pour on half the syrup, let it soak in then pour over the remaining syrup otherwise the blueberries will bleed which doesn't make it taste bad but maybe just not as presentable.
Lime and Blueberry Syrup cake https://www.marcellinaincucina.com/lime-blueberry-syrup-cake/