Algeria's veteran President Abdelaziz Bouteflika has defied the protesters, saying he will run again – but says he will not hold a full term.
In a letter he said if he won April's election, he would oversee a national dialogue leading to new elections that he would not contest.
His decision to seek a fifth term sparked nationwide protests.
Mr. Bouteflika, 82, was rarely seen in public after a stroke in 2013.
On Sunday, new protests saw the midnight deadline for the contestants. At dusk, young people marched again in the capital, Algiers, despite the President's offer.
The campaign manager of Mr Bouteflika has submitted papers on behalf of the ailing president, who is receiving medical treatment in Switzerland.
The electoral commission said that the candidates had to submit them in person, but the Constitutional Council decided that he did not have to be physically present.
How does the idea of the president work?
The proposal was written in the form of a letter to the Algerian people invited on state television.
Should he be re-elected? Mr Bouteflika said he would hold an "inclusive national conference" followed by a vote to elect the next president.
He said he would not run again and appreciated the protests that shook Algeria.
"I listened and heard the warmth of the protesters, and especially the thousands of young people who asked me about the future of our country," said the letter, which was read by an ENTV presenter.
"These young people are understandably concerned about the uncertainties they face, and I have the duty and the will to placate the hearts and minds of my compatriots."
Who else is running?
So far, six other candidates have officially registered, including a retired general, Ali Ghediri, who has promised to bring "change" to Algeria.
Businessman Rachid Nekkaz, who has a considerable Facebook following and is popular with Algeria's young people, announced plans to run, but was considered ineligible.
Instead, his cousin, a car mechanic, also known as Rachid Nekkaz, stepped in and the businessman says he'll be his campaign manager.
Two opposition parties, the Labor Party and the Islamist Movement of the Society for Peace, said they would boycott the election.
Ali Benflis, an important challenger to the previous elections, is also not running.
Are these protests unusual?
Yes, in Algeria there are few public protests, and the protests were the biggest since Mr. Bouteflika came to power 20 years ago.
About ten days ago, demonstrations erupted after Mr Bouteflika announced his resignation.
On Sunday, people went back to the streets of the capital Algiers and other major cities.
Police reportedly used water cannons to destroy students who gathered in Algiers.
There were also demonstrations in France, the former colonial power, where a large Algerian community lives.
"We are not against the president, but he is unconscious, he no longer exists, his generals and those close to him do what they will do behind his back," said a protester of news agency AFP.
Who is Abdelaziz Bouteflika?
Mr Bouteflika came to power in 1999 and is acknowledged to have ended a civil war that killed more than 100,000 people.
Protests against food prices and unemployment broke out during the Arab Spring in 2011, but he reacted by lifting a nearly two-decade-old state of emergency and responded to a major demand from the demonstrators.
After his stroke, he won re-election in an opposition-praised poll and dissolved the country's powerful espionage agency.
Critics say his health means he can not fulfill his duties as president.
Despite the disagreement, Mr Bouteflika still has the tendency to win the elections this year.
Algeria's opposition is split, and Mr Bouteflika won the last presidential elections in 2014, even though he had not conducted any personal election campaigns.