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Royal Canadian Air Force CF-18 Hornets depart after refueling with a KC-135 Stratotanker assigned to the 340th Expeditionary Air Refueling Squadron, October 30, 2014, over Iraq. THE CANADIAN PRESS/HO-U.S. Air Force, Staff Sgt. Perry AstonStaff Sgt. Perry Aston/The Canadian Press

The tragedy of Syria is not just that the country is at war. It is that the country is in a war with more than two sides. To follow the shifting list of players in Syria, and to accurately measure the ever-changing angles and relationships, you need a scorecard and a scientific calculator.

Canada entered this trigonometric conflict in pursuit of a simple, arithmetical objective: rolling back the advance of the so-called Islamic State through targeted air strikes. That sounds clear enough. Given the neighbourhood, it isn't.

Policy-makers in Washington, Ottawa and elsewhere are focusing so much on IS that they may be giving themselves, and the public, a distorted picture of what is happening in the region.

Consider the long and expanding list of actors on the Syrian stage. There is the government of Bashar Assad, which started the war back in the early days of the Arab Spring by meeting peaceful protests with lethal force. Mr. Assad's regime now controls only a relatively small part of the country, but the best estimates are that it has caused most of the war's quarter of a million dead, and is the leading reason why more than half of Syria's population is either internally displaced or refugees.

There is IS, which brutally rules over parts of Syria and also Iraq. It opposes the Assad regime but was also to some extent created by it, as a foil to polarize opinion and frame Mr. Assad as the better alternative. Given that IS now occupies much of the country, that strategy has to some extent failed.

But it has also succeeded. The proof? This week, many Westerners cheered Vladimir Putin's decision to send Russian troops and planes to Syria, because even though he is there to prop up Mr. Assad, the Russian leader claims to be fighting IS. On Thursday, former prime minister Jean Chrétien said: "If Putin wants to help, he should be welcomed." If only it were so simple.

Syria's Rubik's Cube of a civil war includes a multitude of other armed groups of various religious, ideological and ethnic persuasions. Some are related to IS, some oppose it, and many are at various points in between. Some opposition groups are supported by the Saudis and the Gulf states. Some are loosely supported by the United States-led military coalition. Some are the coalition's targets. And on Thursday, Mr. Putin's first air strikes appeared to go after not Islamic State, but other groups fighting Mr. Assad.

Neighbouring Turkey is on the scorecard: It opposes both the Assad regime and IS. But at times, it has attacked some of the regime and IS's most effective opponents.

There is the Lebanon-based, Iranian-backed Hezbollah Shiite militia, which fights for the Assads. There is Iran: It funds the Assad regime, and as we went to press Reuters was reporting that hundreds of Iranian troops were in Syria.

There is also, of course, the U.S.-led coalition, of which Canada is a part. It has planes over Syria and Iraq, hitting IS and related groups. But in Syria, it has no boots and few allies on the ground.

And there is the government of neighbouring Iraq, whose past mistreatment of its own Sunni minority is partly to blame for the strength of IS. The regime in Baghdad is a hall of mirrors; the 2003 American invasion and occupation, and the hundred of billions of American dollars spent, have not turned Iraq into a reliable ally of Washington. The coalition is bombing in Iraq, nominally in support of the Iraqi government – but Baghdad often appears to be closer to Tehran, Damascus and even Moscow. On Thursday, Iraq's prime minister said he would welcome Russian air strikes not only in Syria, but in his country, too.

Beneath all of this political manoeuvring, Syria's multiple wars grind on. The country has become a kind of alternate-plot rerun of what happened a few years ago in Libya. Remember Libya? The West intervened in the civil war, used air power against the government and ensured Moammar Gadhafi's overthrow. It was a military victory, but a political failure. Post-Gadhafi, the country is a fractured, failed state, exporting refugees and violence.

In Syria, the West did the opposite: It did not intervene in the civil war and did not bomb the regime into defeat. Opposite approach, similar result: War rages and Syria is the source of the world's largest refugee crisis.

No matter who wins the Canadian election on Oct. 19, the next government is going to have to talk honestly about what Canada can accomplish in Syria. It is no use invoking the glories of peacekeeping missions of yore; there is no Pearsonian peacekeeping role to be had in Syria. But neither is it right to picture Syria as a simple war against Islamist terrorism, where anyone claiming to be fighting IS should be welcomed as an ally. It may one day be possible for the West to find common ground with Tehran, Moscow, Ankara, Baghdad, the Gulf states, the Saudis and some of the various non-Assad, non-IS militias. But they are primarily in Syria to help themselves – not to help us.

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