CZ-SK-MO 1953 Problem 1

Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $z+\bar z=a\cdot|z|,$ where $a\in\mathbb R$ is given.

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.