First, Russia must make it clear to Iran that it will not be permitted to establish any military bases near the Israeli borders.
Second, it should send a clear message to Hezbollah that it must not be tempted to provoke Israel, as in this regard, Russia cannot prevent Israel from conducting a massive retaliation which could undermine Moscow's strategic interest.
Third, Putin must prevail on Turkey to stop its incursion into Syrian territory and disabuse Erdogan of his quest to subdue the Syrian Kurds, as this will only further aggravate and prolong the conflict in Syria. Putin is convinced that Turkey wants to maintain a permanent presence in Syria, which is a recipe for continuing violence between Turkish forces and the YPG, yet another destabilizing factor.
Fourth, Putin must now seek US involvement in the search for a permanent solution to Syria's civil war. The US remains a dominant regional power and even though Russia is the main power broker in Syria, the US' support remains critical if for no other reason than it has close ties with Israel, and that it might be drawn into in any future war between Israel and Iran/Hezbollah.
The US: Sadly, the Trump administration, which has largely followed Obama's policy toward Syria, is now confronted with a new reality. The US under Trump does not seem to have a clear strategy as to how to deal with the conflict. Moreover, limiting American direct involvement in the conflict only to deter Assad from using chemical weapons against his people, as Trump has done once before, has had little impact on the course of the war and on Assad's behavior, as long as he could count on Russian support.
The current situation in Syria is different for four reasons: 1) President Assad, who was excluded by the Obama administration from being a part of the solution, is assured of remaining president and will certainly be 're-elected' once new elections are held; 2) Iran's direct involvement in Syria's civil war and its ambition to fully entrench itself in the country is a fact that Israel views as a threat against its security; 3) even when the civil war comes to an end, the sectarian conflict and the rivalry for power will continue to haunt the country for years, which is a recipe for destabilization that impacts the US' regional allies; and 4) much of the country lies in ruin and would require tens of billions of dollars for reconstruction, which of necessity requires the US' leadership role to raise the necessary funds.
To prevent miscalculation that could lead to an unintended war between Israel and Iran/Hezbollah, and perhaps the inadvertent involvement of Syria, the US must:
a) Maintain the presence of American troops and advisors that were dispatched to Syria to fight ISIS, and even further augment them to provide the US the leverage it needs to play an important role in the search for a solution, in coordination with Russia.
b) The US ought to restate its commitment to Israel's national security. Additionally, notwithstanding the present strategic defense coordination between two countries, the Trump administration should consider issuing a statement, along the line of its commitment to NATO. The US should state that any major attack on Israel will constitute an attack on the US. This will certainly deter Iran from even contemplating any major hostilities against Israel.
c) Ideally, Trump should focus on amending the Iran deal in cooperation with the other five signatories, and do so through diplomatic channels rather than by issuing an ultimatum to withdraw from it completely by May, which will only heighten regional tensions. Knowing Trump's disdain toward Iran and his characterization of the deal as being 'the worst ever,' he may still withdraw from the deal. At a minimum, however, he should not reinstate the sanctions so that the other signatories will have the opportunity to modify it through negotiations.
Otherwise, the precipitous withdraw from the deal will only unsettle the Iranians and may well prompt them to abandon it altogether, which could potentially lead to regional nuclear proliferation that the US and its allies in the area want to avoid. Moreover, at a time when the US wants to negotiate denuclearization with North Korea, it should not unilaterally revoke the Iran deal and expect the North Koreans to trust the US to live up to its commitments.
The irony is that none of the players involved directly or indirectly in the civil war in Syria want to escalate the conflict by threatening Israel, which will stop short of nothing to protect its national security, especially if the threat is deemed existential. Every party also knows that regardless of how much damage Israel may sustain in such a war, it will emerge victorious while inflicting perhaps unprecedented destruction on its enemies.
In the final analysis, any resolution to a conflict is measured by the prospective losses or gains. There is nothing here to suggest that any of the parties involved foresee a long-term strategic gain that can justify a catastrophic war. A war could erupt as a result of miscalculation, but this can be avoided. Russia in particular and the US must cooperate and lean heavily on their respective clients to prevent such a miscalculation.