A constitutional interpretation, not a political discussion.
I heard someone on a radio news show (a commentator whose name was not familiar to me and whose name I don't now remember) saying that if the House impeaches a president and the Senate convicts that president, removal from office of the president is automatic - but that ineligibility to run for office again is not automatic. "The Senate must decide," he said, "whether removal from office in each case must render the president ineligible for further office. It they make no decision, the president is eligible to run for another term in the next election."
What I am seeking here are comments on a reading of the constitution - not on the merits of the current impeachment inquiry nor on any of the current political players. Nothing more than a reading of the Constitution - is a president who has been impeached and removed from office eligible to run again unless the Senate decides? Since no president has ever been removed from office in this manner, there is no precedent.