THE TWO LARGEST Latin American debtors have been managing their finances with skill and growing indications of success. But both Brazil and Mexico have strong and sophisticated economies. There are other Latin countries where things are going much less well. Peru is a particularly distressing case of a country that was in serious trouble well before the interest rates soared and the debt crisis began. Its debts, in relation to the economy that must support them, are larger than Mexico's. A sense of desperation seems to be taking hold.

After years of solid growth, the Peruvian economy stagnated in the 1970s. Some of it was bad luck, such as the shift in ocean currents that damaged the fishing. Some of the reasons lay closer to home; the country was being run by an incompetent military junta. Income per capita was flat through the decade; since 1980, according to the government, it has fallen by one-fourth.

The strangest and most dismaying sign of internal strain in Peru is the emergence over the past four years of the terrorist organization called the Sendero Luminoso, or Shining Path, drawing its support from the impoverished Indian population of the highlands. They are young people, typically, who have managed to acquire a little education, and who then find that the road to a better life is no longer open to them in a deteriorating economy. The Sendero is a rebellion aimed at everything that belongs to the modern industrial world. More specifically, it is aimed at everything Hispanic. That is why this small organization generates such deep apprehensions in Peru. The country's people today are mainly the descendants of two distinct lines, the Spanish settlers and the Indians who were there before the conquest. The Sendero hints at a revolt of one against the other. More than 3,500 people have been killed so far in the fighting, an astonishing toll since the Sendero has never had more than a couple of thousand members.

The country's elected president, Fernando Belaunde Terry, is coming to the end of his term, and elections are to be held next spring. For reasons familiar to Americans this year, the government is not currently showing much inclination to make difficult decisions on money and deficits. In its agreement with the International Monetary Fund, Peru committed itself to a series of economic targets but is now drifting far off course.

President Belaunde was here in Washington a few weeks ago for talks with the IMF and President Reagan. He got sympathy, but not much more. The view at the IMF is that it will be difficult to help Peru, until Peru can decide how best to help itself. Where larger and richer Latin countries have responded to recession and crisis with vigor, Peru seems to have been overwhelmed and immobilized. But if its government can recover its grip on the economy, Peru will deserve more substantial help than sympathy alone from its friends abroad.