Cairo ( CNN ) -- The next phase of Egypt 's complex elections process continues Monday , with runoff races pitting candidates vying to join the lower house of parliament . The two-day vote -- which continues into Tuesday -- comes exactly a week after Egyptians cast ballots for the first time since last February 's toppling of long-time President Hosni Mubarak . That vote ended with moderate and more conservative Islamist parties together winning big and earning a majority of seats within the new parliament , called the People 's Assembly . Such results appear to mirror recent victories by moderate Islamists in fellow North African nations Morocco and Tunisia . Presidential candidate Amr Moussa , a former Egyptian foreign minister and Arab League secretary-general , told CNN on Sunday that he expected this week 's runoff to change the final mix of parties in the legislative chamber . Still , he suggested that already the announced results should serve as a wake-up call to more liberal and sectarian factions within Egypt . `` The ... final results , I believe , will be more balanced , '' predicted Moussa of the upcoming election . `` ( Still ) , this is a message to the liberal forces that they have to come together and ... mobilize themselves in order to create a strong opposition within the parliament . '' Last week , Egyptians could cast three votes : two for independent candidates and one for a party or coalition . Four independent candidates secured seats , including Amr Hamzawy , once a research director at the Carnegie Endowment for International Peace and a spokesman of the `` Board of Wise Men '' that worked to foster negotiations between the government and anti-government protesters earlier this year . But other positions in parliament remain in limbo because no candidate won clear majorities , leading to this week 's runoff . It 's all part of a multi-step process set up that will shape the makeup of Egypt 's legislature and government going forward . The last step takes place in June with presidential elections , according to military leaders who have in effect ruled the country since Mubarak 's fall . One hot topic that will face the new government is how to handle Egypt 's currently peaceful arrangement with Israel . Under Mubarak and his secular government , Egypt and Israel forged agreements that have resulted in peace and economic cooperation between the two nations . But the successes of Islamists in elections , to date , have raised the specter that major changes , and perhaps rising tensions , could be on the horizon . The relatively moderate Muslim Brotherhood 's Freedom and Justice Party won 40 % in the first round of voting for the lower house of parliament , according to Yousri Abdel Kareem , head of the executive office of the Higher Judicial Election Council . The second highest total , at 20 % , went to members of the Al Noor Salafi Movement , a hard-line Muslim group . Speaking to his nation 's Channel 2 on Sunday , Israeli Defense Minister Ehud Barak called the Egyptian vote results so far `` very troubling . '' Prime Minister Benjamin Netanyahu added Sunday -- while participating in an annual state memorial service for David Ben-Gurion , Israel 's first prime minister -- that both countries would be best served by not significantly altering their relationship . `` I hope that whichever government takes power in Egypt would recognize the importance of maintaining the peace agreement with Israel , '' he said . `` There is an importance in recognizing the peace with Israel , both as a value of its own and as a basis for the financial and security stability of the region . '' But Moussa said that change may be inevitable , given that `` the Middle East of last year is ... gone for good . '' He claimed it should be no surprise that a `` new Arab world '' would want a `` new set of relations , '' stressing that Israel will more likely have positive relationships with its neighbors once it ends its `` occupation '' of Palestinian territories . `` The Israelis must sit now and reflect . Egypt is no ( longer ) the Egypt they knew ( and ) the other neighbors are not the same and will not be the same , '' Moussa said . `` There is a window of opportunity for all of us to solve the problems and move on in a totally new era , including Israel . '' CNN 's Jim Clancy and Guy Azriel and journalist Mohamed Fahmy Fadel contributed to this report .
A two-day runoff starts Monday for remaining seats in Egypt 's parliament . Islamist parties did very well in the first round of voting , held last week . Israeli leaders call results so far `` troubling , '' say Egypt should n't alter its relations . Amr Moussa says it 's expected a `` new Arab world '' wants a `` new set of relations ''